Table of Contents
| 1: A List of Definitions |
| The list of definitions discussed so far in this site |
| 2: Locally Trivial Surjection of Rank r |
| definition of locally trivial surjection of rank \(k\) |
| 3: A List of Propositions |
| The list of propositions discussed so far in this site |
| 4: Connection Depends Only on Section Values on Vector Curve |
| description/proof of that any vector bundle connection depends only on the section values on any vector curve |
| 5: \(C^\infty\) Vectors Bundle |
| definition of \(C^\infty\) vectors bundle of rank \(k\) |
| 6: Surjection |
| A definition of surjection |
| 7: Injection |
| A definition of injection |
| 8: Topology |
| A definition of topology |
| 9: Open Set |
| A definition of open set |
| 10: Closed Set |
| A definition of closed set |
| 11: Neighborhood of Point |
| definition of neighborhood of point |
| 12: Euclidean Topology |
| definition of Euclidean topology |
| 13: Euclidean Topological Space |
| definition of Euclidean topological space |
| 14: Standard Topology for R^n |
| A definition of standard topology for \(\mathbb{R}^n\) |
| 15: Locally Topologically Euclidean Topological Space |
| A definition of locally topologically Euclidean topological space |
| 16: Topological Space |
| A definition of topological space |
| 17: Basis of Topological Space |
| definition of basis of topological space |
| 18: 2nd-Countable Topological Space |
| A definition of 2nd-countable topological space |
| 19: Hausdorff Topological Space |
| A definition of Hausdorff topological space |
| 20: Topological Manifold |
| definition of topological manifold |
| 21: C^infty Manifold |
| A definition of \(C^\infty\) manifold |
| 22: Continuous, Normed Spaces Map |
| definition of continuous, normed vectors spaces map |
| 23: Derivative of Normed Spaces Map |
| A definition of derivative of normed vectors spaces map |
| 24: Derivative of \(C^1\), Euclidean-Normed Euclidean Vectors Spaces Map Is Jacobian |
| description/proof of that derivative of \(C^1\), Euclidean-normed Euclidean vectors spaces map is the Jacobian |
| 25: Chain Rule for Derivative of Compound of C^1, Euclidean-Normed Spaces Maps |
| description/proof of the chain rule for derivative of composition of \(C^1\), Euclidean-normed Euclidean vectors spaces maps |
| 26: Fundamental Theorem of Calculus for Euclidean-Normed Spaces Map |
| description/proof of the fundamental theorem of calculus for \(C^1\), Euclidean-normed spaces map |
| 27: Local Unique Solution Existence for Euclidean-Normed Space ODE |
| A description/proof of the local unique solution existence for Euclidean-normed Euclidean vectors space ordinary differential equation |
| 28: Why Local Solution Existence Does Not Guarantee Global Existence for Euclidean-Normed Space ODE |
| A description of why the local solution existence does not guarantee the global solution existence for Euclidean-normed Euclidean vectors space ODE |
| 29: Contraction Mapping Principle |
| description/proof of the contraction mapping principle |
| 30: Metric Space |
| definition of metric space |
| 31: Lie Algebra |
| definition of Lie algebra |
| 32: General Linear Lie Algebra, \mathfrak{gl} (V) |
| A definition of general linear Lie algebra, \(\mathfrak{gl} (V)\) |
| 33: Normed Vectors Space |
| A definition of normed vectors space |
| 34: Inner Product on Real or Complex Vectors Space |
| A definition of inner product on real or complex vectors space |
| 35: Cauchy-Schwarz Inequality for Real or Complex Inner-Producted Vectors Space |
| description/proof of the Cauchy-Schwarz inequality for real or complex inner-producted vectors space |
| 36: Inner Product on Real or Complex Vectors Space Induces Norm |
| description/proof of that inner product on real or complex vectors space induces norm |
| 37: Isomorphism Between Tangent Space of General Linear Group at Identity and General Linear Lie Algebra |
| description/proof of that tangent vectors space of general linear group of finite-dimensional real vectors space at identity is 'vectors spaces - linear morphisms' isomorphic to general linear Lie algebra |
| 38: Derivative of Real-1-Parameter Family of Vectors |
| definition of derivative of real-1-parameter family of vectors in finite-dimensional real vectors space |
| 39: Norm on Real or Complex Vectors Space |
| definition of norm on real or complex vectors space |
| 40: Metric |
| definition of metric |
| 41: Continuous Map at Point |
| A definition of map continuous at point |
| 42: Continuous Map |
| A definition of continuous map |
| 43: Local Criterion for Openness |
| description/proof of local criterion for openness |
| 44: Topological Path |
| definition of topological path |
| 45: Path-Connected Topological Space |
| A definition of path-connected topological space |
| 46: Connected Topological Space |
| A definition of connected topological space |
| 47: Connected Topological Manifold Is Path-Connected |
| description/proof of that connected topological manifold is path-connected |
| 48: General Linear Group of Vectors Space |
| definition of general linear group of vectors space |
| 49: Limit Condition Can Be Substituted with With-Equal Conditions |
| description/proof of that limit condition of normed vectors spaces map can be substituted with with-equal conditions |
| 50: Residue of Derivative of Normed-Spaces Map Is Differentiable at Point If ..., and the Derivative Is ... |
| A description/proof of that residue of derivative of normed vectors spaces map is differentiable at point of 2nd argument if original map is differentiable at corresponding point with derivative as minus original map derivative at 1st argument point plus original map derivative at corresponding point |
| 51: Inverse Theorem for Euclidean-Normed Spaces Map |
| A description/proof of the inverse theorem for Euclidean-normed spaces map |
| 52: Existence of Lie Group Neighborhood Whose Any Point Can Be Connected with Center by Left-Invariant Vectors Field Integral Curve |
| A description/proof of existence of Lie group neighborhood whose any point can be connected with center by left-invariant vectors field integral curve |
| 53: 2 Points on Connected Lie Group Can Be Connected by Finite Left-Invariant Vectors Field Integral Curve Segments |
| A description/proof of that 2 points on connected Lie group can be connected by finite left-invariant vectors field integral curve segments |
| 54: Point on Connected Lie Group Can Be Expressed as Finite Product of Exponential Maps |
| A description/proof of that point on connected Lie group can be expressed as finite product of exponential maps |
| 55: Left-Invariant Vectors Field on Lie Group Is C^infty |
| A description/proof of that left-invariant vectors field on Lie group is \(C^\infty\) |
| 56: Vectors Field Is C^\infty If and Only If Operation Result on Any C^\infty Function Is C^\infty |
| A description/proof of that vectors field is \(C^\infty\) if and only if operation result on any \(C^\infty\) function is \(C^\infty\) |
| 57: Germ of C^k Functions at Point |
| A definition of germ of \(C^k\) functions at point, \(C^k_p (M)\) |
| 58: Derivation at Point of C^k Functions |
| A definition of derivation at point of \(C^k\) functions |
| 59: Tangent Vector |
| A definition of tangent vector |
| 60: Directional Derivative |
| A definition of directional derivative |
| 61: Equivalence Between Derivation at Point of C^1 Functions and Directional Derivative |
| A description/proof of equivalence between derivation at point of \(C^1\) functions and directional derivative |
| 62: Map |
| A definition of map |
| 63: Homeomorphism |
| A definition of homeomorphism |
| 64: Homeomorphic Topological Manifolds Can Have Equivalent Atlases |
| A description/proof of that homeomorphic topological manifolds can have equivalent atlases |
| 65: Map Preimage of Whole Codomain Is Whole Domain |
| A description/proof of that map preimage of whole codomain is whole domain |
| 66: Map Preimage of Codomain Minus Set Is Domain Minus Preimage of Set |
| A description/proof of that map preimage of codomain minus set is domain minus preimage of set |
| 67: Continuous Map Preimage of Closed Set Is Closed Set |
| A description/proof of that continuous map preimage of closed set is closed set |
| 68: Preimage of Non-Zero Determinants of Matrix of Continuous Functions Is Open |
| A description/proof of that preimage of non-zero determinants of matrix of continuous functions is open |
| 69: Induced Map from Domain Quotient of Continuous Map Is Continuous |
| A description/proof of that induced map from domain quotient of continuous map is continuous |
| 70: Subset of R^{d-k} Is Open If the Product of R^k and Subset Is Open |
| A description/proof of that subset of \(R^{d-k}\) is open if the product of \(R^k\) and subset is open |
| 71: For \(C^\infty\) Function on Open Neighborhood, There Exists \(C^\infty\) Function on \(C^\infty\) Manifold with Boundary That Equals Function on Possibly Smaller Neighborhood |
| description/proof of that for \(C^\infty\) function on open neighborhood, there exists \(C^\infty\) function on \(C^\infty\) manifold with boundary that equals function on possibly smaller neighborhood |
| 72: Preimage by Product Map Is Product of Preimages by Component Maps |
| A description/proof of that preimage by product map is product of preimages by component maps |
| 73: Product Map of Continuous Maps Is Continuous |
| A description/proof of that product map of continuous maps is continuous |
| 74: Some Para-Product Maps of Continuous Maps Are Continuous |
| A description/proof of that some para-product maps of continuous maps are continuous |
| 75: Map Image of Union of Sets Is Union of Map Images of Sets |
| A description/proof of that map image of union of sets is union of map images of sets |
| 76: Map Preimage of Union of Sets Is Union of Map Preimages of Sets |
| A description/proof of that map preimage of union of sets is union of map preimages of sets |
| 77: Map Image of Intersection of Sets Is Not Necessarily Intersection of Map Images of Sets |
| A description/proof of that map image of intersection of sets is not necessarily intersection of map images of sets |
| 78: Map Preimage of Intersection of Sets Is Intersection of Map Preimages of Sets |
| A description/proof of that map preimage of intersection of sets is intersection of map preimages of sets |
| 79: Structure |
| definition of structure |
| 80: %Structure Kind Name% Homomorphism |
| definition of %structure kind name% homomorphism |
| 81: Category |
| definition of category |
| 82: Morphism |
| definition of morphism |
| 83: Covariant Functor |
| definition of covariant functor |
| 84: Contravariant Functor |
| definition of contravariant functor |
| 85: Abelian Group |
| A definition of Abelian group |
| 86: Monoid |
| A definition of monoid |
| 87: Group |
| A definition of group |
| 88: Unique Existence of Monoid Identity Element |
| A description/proof of unique existence of monoid identity element |
| 89: Ring |
| A definition of ring |
| 90: Ideal of Ring |
| definition of ideal of ring |
| 91: Quotient Ring of Ring |
| definition of quotient ring of ring by ideal |
| 92: Left R-Module |
| A definition of left R-module |
| 93: Wedge Product |
| A definition of wedge product |
| 94: How Wedge Product as an Equivalence Class of Elements of Tensor Algebra Is Related with the Tensor Products Construct |
| A description of how wedge product as an equivalence class of elements of tensor algebra is related with the tensor products construct |
| 95: Simplex Is Homeomorphic to Same-Dimensional Closed Ball |
| description/proof of that standard simplex is homeomorphic to same-dimensional closed ball |
| 96: For Compact C^\infty Manifold, Sequence of Points Has Convergent Subsequence |
| A description/proof of that for compact \(C^\infty\) manifold, sequence of points has convergent subsequence |
| 97: Image of Continuous Map from Compact Topological Space to \mathbb{R} Euclidean Topological Space Has Minimum and Maximum |
| A description/proof of that image of continuous map from compact topological space to \(\mathbb{R}\) Euclidean Topological Space has minimum and maximum |
| 98: Intersection or Finite Union of Closed Sets Is Closed |
| A description/proof of that intersection or finite union of closed sets is closed |
| 99: Intersection of Complements of Subsets Is Complement of Union of Subsets |
| description/proof of that intersection of complements of subsets is complement of union of subsets |
| 100: Union of Complements of Subsets Is Complement of Intersection of Subsets |
| description/proof of that union of complements of subsets is complement of intersection of subsets |
| 101: Subspace Topology |
| definition of subspace topology of subset of topological space |
| 102: Chart on Regular Submanifold Is Extension of Adapting Chart |
| A description/proof of that chart on regular submanifold is extension of adapting chart |
| 103: Regular Submanifold of Regular Submanifold Is Regular Submanifold of Base C^\infty Manifold of Specific Codimension |
| A description/proof of that regular submanifold of regular submanifold is regular submanifold of base \(C^\infty\) manifold of specific codimension |
| 104: Intersection of 2 Transversal Regular Submanifolds of C^\infty Manifold Is Regular Submanifold of Specific Codimension |
| A description/proof of that intersection of 2 transversal regular submanifolds of \(C^\infty\) manifold is regular submanifold of specific codimension |
| 105: Subset of Open Topological Subspace Is Open on Subspace Iff It Is Open on Base Space |
| A description/proof of that subset of open topological subspace is open on subspace iff it is open on base space |
| 106: Subset of Not-Necessarily-Open Topological Subspace Is Open on Subspace If It Is Open on Basespace |
| description/proof of that subset of not-necessarily-open topological subspace is open on subspace if it is open on basespace |
| 107: Basis Determines Topology |
| A description/proof of that basis determines topology |
| 108: Linear Surjection from Finite Dimensional Vectors Space to Same Dimensional Vectors Space Is 'Vectors Spaces - Linear Morphisms' Isomorphism |
| description/proof of that linear surjection from finite-dimensional vectors space to same-dimensional vectors space is 'vectors spaces - linear morphisms' isomorphism |
| 109: Map Image of Point Is On Subset Iff Point Is on Preimage of Subset |
| A description/proof of that map image of point is on subset iff point is on preimage of subset |
| 110: Point Is on Map Image of Subset if Preimage of Point Is Contained in Subset, but Not Only if |
| A description/proof of that point is on map image of subset if preimage of point is contained in subset, but not only if |
| 111: Composition of Map After Preimage Is Contained in Argument Set |
| A description/proof of that composition of map after preimage is contained in argument set |
| 112: Map Image of Subset Is Contained in Subset iff Subset Is Contained in Preimage of Subset |
| A description/proof of that map image of subset is contained in subset iff subset is contained in preimage of subset |
| 113: Preimage Under Domain-Restricted Map Is Intersection of Preimage Under Original Map and Restricted Domain |
| A description/proof of that preimage under domain-restricted map is intersection of preimage under original map and restricted domain |
| 114: Subset of Subspace of Adjunction Topological Space Is Open Iff Projections of Preimage of Subset Are Open with Condition |
| A description/proof of that subset of subspace of adjunction topological space is open iff projections of preimage of subset are open with condition. |
| 115: Reverse of Tietze Extension Theorem |
| A description/proof of reverse of Tietze extension theorem |
| 116: Some Properties Concerning Adjunction Topological Space When Inclusion to Attaching-Origin Space from Subset Is Closed Embedding |
| A description/proof of some properties about adjunction topological space when inclusion to attaching-origin space from subset is closed embedding |
| 117: Linear Range of Finite-Dimensional Vectors Space Is Vectors Space |
| description/proof of that linear range of finite-dimensional vectors space is vectors space |
| 118: %Category Name% Isomorphism |
| definition of %category name% isomorphism |
| 119: For Linear Map from Finite Dimensional Vectors Space, There Is Domain Subspace That Is 'Vectors Spaces - Linear Morphisms' Isomorphic to Image by Restriction of Map |
| A description/proof of that for linear map from finite dimensional vectors space, there is domain subspace that is 'vectors Spaces - linear morphisms' isomorphic to image by restriction of map |
| 120: Curves on Manifold as the C^\infty Right Actions of Curves That Represent Same Vector on Lie Group Represent Same Vector |
| A description/proof of that curves on manifold as the \(C^\infty\) right actions of curves that represent same vector on Lie group represent same vector |
| 121: Parameterized Family of Vectors and Curve Induced by C^\infty Right Action of Lie Group Represent Same Vector If . . . |
| A description/proof of that parameterized family of vectors and curve induced by \(C^\infty\) right action of Lie group represent same vector if . . . |
| 122: Finite Dimensional Vectors Spaces Related by Linear Bijection Are of Same Dimension |
| A description/proof of that finite dimensional vectors spaces related by linear bijection are of same dimension |
| 123: Absolute Difference Between Complex Numbers Is or Above Difference Between Absolute Differences with Additional Complex Number |
| A description/proof of that absolute difference between complex numbers is or above difference between absolute differences with additional complex number |
| 124: Injective Map Image of Intersection of Sets Is Intersection of Map Images of Sets |
| A description/proof of that injective map image of intersection of sets is intersection of map images of sets |
| 125: Finite Dimensional Real Vectors Space Topology Defined Based on Coordinates Space Does Not Depend on Choice of Basis |
| A description/proof of that finite dimensional real vectors space topology defined based on coordinates space does not depend on choice of basis |
| 126: 'Real Vectors Spaces-Linear Morphisms' Isomorphism Between Topological Spaces with Coordinates Topologies Is Homeomorphic |
| A description/proof of that 'real vectors spaces-linear morphisms' isomorphism between topological spaces with coordinates topologies is homeomorphic |
| 127: Normal Topological Space |
| A definition of normal topological space |
| 128: Closure of Subset |
| A definition of closure of subset of topological space |
| 129: Topological Space Is Normal Iff for Closed Set and Its Containing Open Set There Is Closed-Set-Containing Open Set Whose ~ |
| A description/proof of that topological space is normal iff for closed set and its containing open set there is closed-set-containing open set whose ~ |
| 130: Equivalence of Map Continuousness in Topological Sense and in Norm Sense for Coordinates Functions |
| A description/proof of equivalence of map continuousness in topological sense and in norm sense for coordinates functions |
| 131: Euclidean Topological Space Nested in Euclidean Topological Space Is Topological Subspace |
| A description/proof of that Euclidean topological space nested in Euclidean topological space is topological subspace |
| 132: For Map Between Topological Spaces and Domain Point, if There Are Superspaces of Domain and Codomain, Open Neighborhoods of Point and of Point Image on Superspaces, and Continuous Map from Domain Open Neighborhood into Codomain neighborhood That Is Restricted to Original Map on Intersection of Domain Neighborhood and Original Domain, Original Map Is Continuous at Point |
| description/proof of that for map between topological spaces and domain point, if there are superspaces of domain and codomain, open neighborhoods of point and of point image on superspaces, and continuous map from domain neighborhood into codomain neighborhood that is restricted to original map on intersection of domain neighborhood and original domain, original map is continuous at point |
| 133: Criteria for Collection of Open Sets to Be Basis |
| A description/proof of criteria for collection of open sets to be basis |
| 134: C^1 Map from Open Set on Euclidean Normed C^\infty Manifold to Euclidean Normed C^\infty Manifold Locally Satisfies Lipschitz Condition |
| A description/proof of that \(C^1\) map from open set on Euclidean normed \(C^\infty\) manifold to Euclidean normed \(C^\infty\) manifold locally satisfies Lipschitz condition |
| 135: Area of Hyperrectangle Can Be Approximated by Area of Covering Finite Number Hypersquares to Any Precision |
| A description/proof of that area of hyperrectangle can be approximated by area of covering finite number hypersquares to any precision |
| 136: Area on Euclidean Metric Space Can Be Measured Using Only Hypersquares, Instead of Hyperrectangles |
| A description/proof of that area on Euclidean metric space can be measured using only hypersquares, instead of hyperrectangles |
| 137: From Euclidean Normed Topological Space into Equal or Higher Dimensional Euclidean Normed Topological Space Lipschitz Condition Satisfying Map Image of Measure 0 Subset Is Measure 0 |
| A description/proof of that from Euclidean normed topological space into equal or higher dimensional Euclidean normed topological space Lipschitz condition satisfying map image of measure 0 subset is measure 0 |
| 138: From Convex Open Set Whose Closure Is Bounded on Euclidean Normed C^\infty Manifold into Equal or Higher Dimensional Euclidean Normed C^\infty Manifold Polynomial Map Image of Measure 0 Subset Is Measure 0 |
| A description/proof of that from convex open set whose closure is bounded on Euclidean normed \(C^\infty\) manifold into equal or higher dimensional Euclidean normed \(C^\infty\) manifold polynomial map image of measure 0 subset is measure 0 |
| 139: Open Set Complement of Measure 0 Subset Is Dense |
| A description/proof of that open set complement of measure 0 subset is dense |
| 140: Open Set Minus Closed Set Is Open |
| A description/proof of that open set minus closed set is open |
| 141: For Topological Space, Intersection of Basis and Subspace Is Basis for Subspace |
| A description/proof of that for topological space, intersection of basis and subspace is basis for subspace |
| 142: Closure of Difference of Subsets Is Not Necessarily Difference of Closures of Subsets, But Is Contained in Closure of Minuend |
| A description/proof of that closure of difference of subsets is not necessarily difference of closures of subsets, but is contained in closure of minuend |
| 143: Compact Topological Space Has Accumulation Point of Subset with Infinite Points |
| A description/proof of that compact topological space has accumulation point of subset with infinite points |
| 144: Closed Discrete Subspace of Compact Topological Space Has Only Finite Points |
| A description/proof of that closed discrete subspace of compact topological space has only finite points |
| 145: Finite-Open-Sets-Sequence-Connected Pair of Open Sets |
| A definition of finite-open-sets-sequence-connected pair of open sets |
| 146: Pair of Open Sets of Connected Topological Space Is Finite-Open-Sets-Sequence-Connected |
| A description/proof of that pair of open sets of connected topological space is finite-open-sets-sequence-connected |
| 147: Pair of Elements of Open Cover of Connected Topological Space Is Finite-Open-Sets-Sequence-Connected Via Cover Elements |
| A description/proof of that pair of elements of open cover of connected topological space is finite-open-sets-sequence-connected via cover elements |
| 148: Regular Topological Space |
| A definition of regular topological space |
| 149: Map That Is Anywhere Locally Constant on Connected Topological Space Is Globally Constant |
| A description/proof of that map that is anywhere locally constant on connected topological space is globally constant |
| 150: For Regular Topological Space, Neighborhood of Point Contains Closed Neighborhood |
| A description/proof of that for regular topological space, neighborhood of point contains closed neighborhood |
| 151: Identity Map with Domain and Codomain Having Different Topologies Is Continuous iff Domain Is Finer than Codomain |
| A description/proof of that identity map with domain and codomain having different topologies is continuous iff domain is finer than codomain |
| 152: For Metric Space, Difference of Distances of 2 Points from Subset Is Equal to or Less Than Distance Between Points |
| A description/proof of that for metric space, difference of distances of 2 points from subset is equal to or less than distance between points |
| 153: Directed Set |
| A definition of directed set |
| 154: Net with Directed Index Set |
| A definition of net with directed indices set |
| 155: Convergence of Net with Directed Index Set |
| definition of convergence of net with directed index set |
| 156: For Metric Space, 1 Point Subset Is Closed |
| A description/proof of that for metric space, 1 point subset is closed |
| 157: For Hausdorff Topological Space, Net with Directed Index Set Can Have Only 1 Convergence |
| A description/proof of that for Hausdorff topological space, net with directed index set can have only 1 convergence |
| 158: Accumulation Value of Net with Directed Index Set |
| A definition of accumulation value of net with directed index set |
| 159: Final Map Between Directed Sets |
| A definition of final map between directed sets |
| 160: Subnet of Net with Directed Index Set |
| A definition of subnet of net with directed index set |
| 161: Accumulation Value of Net with Directed Index Set Is Convergence of Subnet |
| A description/proof of that accumulation value of net with directed index set is convergence of subnet |
| 162: \(C^\infty\) Embedding |
| definition of \(C^\infty\) embedding |
| 163: Continuous Embedding |
| A definition of continuous embedding |
| 164: Open Set on Open Topological Subspace Is Open on Base Space |
| A description/proof of that open set on open topological subspace is open on base space |
| 165: Closed Set on Closed Topological Subspace Is Closed on Base Space |
| A description/proof of that closed set on closed topological subspace is closed on base space |
| 166: Map Between Topological Spaces Is Continuous if Domain Restriction of Map to Each Open Set of Open Cover is Continuous |
| A description/proof of that map between topological spaces is continuous if domain restriction of map to each open set of open cover is continuous |
| 167: Map Between Topological Spaces Is Continuous if Domain Restriction of Map to Each Closed Set of Finite Closed Cover is Continuous |
| A description/proof of that map between topological spaces is continuous if domain restriction of map to each closed set of finite closed cover is continuous |
| 168: If Preimage of Closed Set Under Topological Spaces Map Is Closed, Map Is Continuous |
| A description/proof of that if preimage of closed set under topological spaces map is closed, map is continuous |
| 169: Composition of Map After Preimage Is Identical Iff Argument Set Is Subset of Map Image |
| description/proof of that composition of map after preimage is identical iff argument set is subset of map range |
| 170: Composition of Preimage After Map of Subset Is Identical If Map Is Injective with Respect to Argument Set Image |
| A description/proof of that composition of preimage after map of subset is identical if map is injective with respect to argument set image |
| 171: Composition of Preimage After Map of Subset Is Identical Iff It Is Contained in Argument Set |
| A description/proof of that composition of preimage after map of subset is identical iff it is contained in argument set |
| 172: Restriction of Continuous Map on Domain and Codomain Is Continuous |
| A description/proof of that restriction of continuous map on domain and codomain is continuous |
| 173: Quotient Map |
| A definition of quotient map |
| 174: Universal Property of Continuous Embedding |
| A description/proof of universal property of continuous embedding |
| 175: Universal Property of Quotient Map |
| A description/proof of universal property of quotient map |
| 176: When Image of Point Is on Image of Subset, Point Is on Subset if Map Is Injective with Respect to Image of Subset |
| A description/proof of that when image of point is on image of subset, point is on subset if map is injective with respect to image of subset |
| 177: Topological Sum |
| A definition of topological sum |
| 178: Adjunction Topological Space Obtained by Attaching Topological Space via Map to Topological Space |
| A definition of adjunction topological space obtained by attaching topological space via map to topological space |
| 179: Quotient Topology on Set with Respect to Map |
| A definition of quotient topology on set with respect to map |
| 180: Quotient Topology Is Sole Finest Topology That Makes Map Continuous |
| A description/proof of that quotient topology is sole finest topology that makes map continuous |
| 181: Map of Quotient Topology Is Quotient Map |
| A description/proof of that map of quotient topology is quotient map |
| 182: Subset of Quotient Topological Space Is Closed iff Preimage of Subset Under Quotient Map Is Closed |
| A description/proof of that subset of quotient topological space is closed iff preimage of subset under quotient map is closed |
| 183: Map Preimages of Disjoint Subsets Are Disjoint |
| A description/proof of that map preimages of disjoint subsets are disjoint |
| 184: If Union of Disjoint Subsets Is Open, Each Subset Is Not Necessarily Open |
| A description/proof of that if union of disjoint subsets is open, each subset is not necessarily open |
| 185: If Union of Disjoint Subsets Is Closed, Each Subset Is Not Necessarily Closed |
| A description/proof of that if union of disjoint subsets is closed, each subset is not necessarily closed |
| 186: Maps Composition Preimage Is Composition of Map Preimages in Reverse Order |
| A description/proof of that maps composition preimage is composition of map preimages in reverse order |
| 187: For Disjoint Subset and Open Set, Closure of Subset and Open Set Are Disjoint |
| A description/proof of that for disjoint subset and open set, closure of subset and open set are disjoint |
| 188: Closure of Subset Is Union of Subset and Accumulation Points Set of Subset |
| A description/proof of that closure of subset is union of subset and accumulation points set of subset |
| 189: Local Characterization of Closure: Point Is on Closure of Subset iff Its Every Neighborhood Intersects Subset |
| description/proof of that local characterization of closure: point is on closure of subset iff every neighborhood of point intersects subset |
| 190: Subset Is Contained in Map Preimage of Image of Subset |
| A description/proof of that subset is contained in map preimage of image of subset |
| 191: Map Image of Intersection of Sets Is Contained in Intersection of Map Images of Sets |
| A description/proof of that map image of intersection of sets is contained in intersection of map images of sets |
| 192: 2 Metrics with Condition with Each Other Define Same Topology |
| A description/proof of that 2 metrics with condition with each other define same topology |
| 193: For Adjunction Topological Space, Canonical Map from Attaching-Destination Space to Adjunction Space Is Continuous Embedding |
| A description/proof of that for adjunction topological space, canonical map from attaching-destination space to adjunction space is continuous embedding |
| 194: Set of Subsets with Whole Set and Empty Set Constitutes Subbasis |
| A description/proof of that set of subsets with whole set and empty set constitutes subbasis |
| 195: Set of Neighborhood Bases at All Points Determines Topology |
| A description/proof of that set of neighborhood bases at all points determines topology |
| 196: Open Set Intersects Subset if It Intersects Closure of Subset |
| A description/proof of that open set intersects subset if it intersects closure of subset |
| 197: Connected Topological Component |
| A definition of connected topological component |
| 198: Topological Connected-Ness of 2 Points |
| A definition of topological connected-ness of 2 points |
| 199: Topological Path-Connected-Ness of 2 Points |
| A definition of topological path-connected-ness of 2 points |
| 200: Topological Path-Connected-ness of 2 Points Is Equivalence Relation |
| A description/proof of that topological path-connected-ness of 2 points is equivalence relation |
| 201: Path-Connected Topological Component |
| A definition of path-connected topological component |
| 202: Connected Topological Component Is Exactly Connected Topological Subspace That Cannot Be Made Larger |
| A description/proof of that connected topological component is exactly connected topological subspace that cannot be made larger |
| 203: Topological Connected-ness of 2 Points Is Equivalence Relation |
| A description/proof of that topological connected-ness of 2 points is equivalence relation |
| 204: Subspace That Contains Connected Subspace and Is Contained in Closure of Connected Subspace Is Connected |
| A description/proof of that subspace that contains connected subspace and is contained in closure of connected subspace is connected |
| 205: Expansion of Continuous Map on Codomain Is Continuous |
| A description/proof of that expansion of continuous map on codomain is continuous |
| 206: 2 Points Are Topologically Path-Connected iff There Is Path That Connects 2 Points |
| A description/proof of that 2 points are topologically path-connected iff there is path that connects 2 points |
| 207: 2 Points That Are Path-Connected on Topological Subspace Are Path-Connected on Larger Subspace |
| A description/proof of that 2 points that are path-connected on topological subspace are path-connected on larger subspace |
| 208: Union of Path-Connected Subspaces Is Path-Connected if Subspace of Point from Each Subspace Is Path-Connected |
| A description/proof of that union of path-connected subspaces is path-connected if subspace of point from each subspace is path-connected |
| 209: Path-Connected Topological Component Is Exactly Path-Connected Topological Subspace That Cannot Be Made Larger |
| A description/proof of that path-connected topological component is exactly path-connected topological subspace that cannot be made larger |
| 210: Locally Path-Connected Topological Space |
| A definition of locally path-connected topological space |
| 211: Locally Connected Topological Space |
| A definition of locally connected topological space |
| 212: Connected Component Is Open on Locally Connected Topological Space |
| A description/proof of that connected component is open on locally connected topological space |
| 213: Connected Component Is Closed |
| A description/proof of that connected component is closed |
| 214: Topological Space Is Connected iff Its Open and Closed Subsets Are Only It and Empty Set |
| A description/proof of that topological space is connected iff its open and closed subsets are only it and empty set |
| 215: Subset on Topological Subspace Is Closed iff There Is Closed Set on Base Space Whose Intersection with Subspace Is Subset |
| A description/proof of that subset on topological subspace is closed iff there is closed set on base space whose intersection with subspace is subset |
| 216: In Nest of Topological Subspaces, Connected-ness of Subspace Does Not Depend on Superspace |
| A description/proof of that in nest of topological subspaces, connected-ness of subspace does not depend on superspace |
| 217: Union of 2 Connected Subspaces Is Connected if Each Neighborhood of Point on Subspace Contains Point of Other Subspace |
| A description/proof of that union of 2 connected subspaces is connected if each neighborhood of point on subspace contains point of other subspace |
| 218: Product of Finite Number of Connected Topological Spaces Is Connected |
| A description/proof of that product of finite number of connected topological spaces is connected |
| 219: Path-Connected Topological Component Is Open and Closed on Locally Path-Connected Topological Space |
| A description/proof of that path-connected topological component is open and closed on locally path-connected topological space |
| 220: Neighborhood Basis at Point |
| A definition of neighborhood basis at point |
| 221: Set of Vectors Space Homomorphisms Constitutes Vectors Space |
| A description/proof of that set of vectors space homomorphisms constitutes vectors space |
| 222: Double Dual of Finite Dimensional Real Vectors Space Is 'Vectors Spaces - Linear Morphisms' Isomorphic to Vectors Space |
| A description/proof of that double dual of finite dimensional real vectors space is 'vectors spaces - linear morphisms' isomorphic to vectors space |
| 223: Dual of Finite Dimensional Real Vectors Space Constitutes Same Dimensional Vectors Space |
| A description/proof of that dual of finite dimensional real vectors space constitutes same dimensional vectors space |
| 224: In Nest of Topological Subspaces, Openness of Subset on Subspace Does Not Depend on Superspace |
| A description/proof of that in nest of topological subspaces, openness of subset on subspace does not depend on superspace |
| 225: Open Sets Whose Complements Are Finite and Empty Set Is Topology |
| A description/proof of that open sets whose complements are finite and empty set is topology |
| 226: For Set Plus Set as an Element, Open Sets That Are Subsets of Set and Subsets Whose Complements Are Finite Is Topology |
| A description/proof of that for set plus set as an element, open sets that are subsets of set and subsets whose complements are finite is topology |
| 227: Stereographic Projection Is Homeomorphism |
| A description/proof of that stereographic projection is homeomorphism |
| 228: Set of Subsets Around Each Point with Conditions Generates Unique Topology with Each Set Being Neighborhood Basis |
| A description/proof of that set of subsets around each point with conditions generates unique topology with each set being neighborhood basis |
| 229: Open Set on Euclidean Topological Space Has Rational Point |
| A description/proof of that open set on Euclidean topological space has rational point |
| 230: For Euclidean Topological Space, Set of All Open Balls with Rational Centers and Rational Radii Is Basis |
| A description/proof of that for Euclidean topological space, set of all open balls with rational centers and rational radii is basis |
| 231: Euclidean Topological Space Is 2nd Countable |
| A description/proof of that Euclidean topological space is 2nd countable |
| 232: Subspace of 2nd Countable Topological Space Is 2nd Countable |
| A description/proof of that subspace of 2nd countable topological space is 2nd countable |
| 233: For Bijection, Preimage of Subset Under Inverse of Map Is Image of Subset Under Map |
| A description/proof of that for bijection, preimage of subset under inverse of map is image of subset under map |
| 234: For Injective Closed Map Between Topological Spaces, Inverse of Codomain-Restricted-to-Range Map Is Continuous |
| A description/proof of that for injective closed map between topological spaces, inverse of codomain-restricted-to-range map is continuous |
| 235: Disjoint Union Topology |
| A definition of disjoint union topology |
| 236: For Disjoint Union Topological Space, Inclusion from Constituent Topological Space to Disjoint Topological Space Is Continuous |
| A description/proof of that for disjoint union topological space, inclusion from constituent topological space to disjoint topological space is continuous |
| 237: Product Topology |
| definition of product topology |
| 238: Product of Closed Sets Is Closed in Product Topology |
| A description/proof of that product of closed sets is closed in product topology |
| 239: Disjoint Union of Closed Sets Is Closed in Disjoint Union Topology |
| A description/proof of that disjoint union of closed sets is closed in disjoint union topology |
| 240: Some Facts about Separating Possibly-Higher-than-2-Dimensional Matrix into Symmetric Part and Antisymmetric Part w.r.t. Indices Pair |
| A description/proof of that some facts about separating possibly-higher-than-2-dimensional matrix into symmetric part and antisymmetric part w.r.t. indices pair |
| 241: Disjoint Union of Complements Is Disjoint Union of Whole Sets Minus Disjoint Union of Subsets |
| A description/proof of that disjoint union of complements is disjoint union of whole sets minus disjoint union of subsets |
| 242: Product of Any Complements Is Product of Whole Sets Minus Union of Products of Whole Sets 1 of Which Is Replaced with Subset for Each Product |
| A description/proof of that product of any complements is product of whole sets minus union of products of whole sets 1 of which is replaced with subset for each product |
| 243: Difference of Map Images of Subsets Is Contained in Map Image of Difference of Subsets |
| A description/proof of that difference of map images of subsets is contained in map image of difference of subsets |
| 244: Difference of Map Images of Subsets Is Map Image of Difference of Subsets if Map Is Injective |
| A description/proof of that difference of map images of subsets is map image of difference of subsets if map is injective |
| 245: For Quotient Map, Codomain Subset Is Closed if Preimage of Subset Is Closed |
| A description/proof of that for quotient map, codomain subset is closed if preimage of subset is closed |
| 246: Complement of Product of Subsets Is Union of Products of Whole Sets 1 of Which Is Replaced with Complement of Subset |
| A description/proof of that complement of product of subsets is union of products of whole sets 1 of which is replaced with complement of subset |
| 247: For Quotient Map, Induced Map from Quotient Space of Domain by Map to Codomain Is Continuous |
| A description/proof of that for quotient map, induced map from quotient space of domain by map to codomain is continuous |
| 248: For Map Between Real Closed Intervals and Graph of Map as Topological Subspace, Subset Such That Value Is Larger or Smaller Than Independent Variable Is Open |
| A description/proof of that for map between real closed intervals and graph of map as topological subspace, subset such that value is larger or smaller than independent variable is open |
| 249: Subgroup of Abelian Additive Group Is Retract of Group Iff There Is Another Subgroup Such That Group is Sum of Subgroups |
| A description/proof of that subgroup of Abelian additive group is retract of group iff there is another subgroup such that group is sum of subgroups |
| 250: There Is No Set That Contains All Sets |
| A description/proof of that there is no set that contains all sets |
| 251: Collection of Sets That Are of Non-0 Cardinality Is Not Set |
| A description/proof of that collection of sets that are of non-zero cardinality is not set |
| 252: Products of Sets Are Associative in 'Sets - Map Morphisms' Isomorphism Sense |
| A description/proof of that products of sets are associative in 'sets - map morphisms' isomorphism sense |
| 253: Multiplications of Cardinalities of Sets Are Associative |
| A description/proof of that multiplications of cardinalities of sets are associative |
| 254: 'Natural Number'-th Power of Cardinality of Set Is That Times Multiplication of Cardinality |
| A description/proof of that 'natural number'-th power of cardinality of set is that times multiplication of cardinality |
| 255: Cardinality of Multiple Times Multiplication of Set Is That Times Multiplication of Cardinality of Set |
| A description/proof of that cardinality of multiple times multiplication of set is that times multiplication of cardinality of set |
| 256: From Natural Number to Countable Set Functions Set Is Countable |
| A description/proof of that from natural number to countable set functions set is countable |
| 257: For Nonempty Set with Partial Ordering with No Minimal Element, There Is Function from Natural Numbers Set to Set, for Which Image of Number Is Larger than Image of Next Number |
| A description/proof of that for nonempty set with partial ordering with no minimal element, there is function from natural numbers set to set, for which image of number is larger than image of next number |
| 258: Part of Set Is Subset if There Is Formula That Determines Each Element of Set to Be in or out of Part |
| A description/proof of that part of set is subset if there is formula that determines each element of set to be in or out of part |
| 259: Formula That Uniquely Maps Each Element of Set into Set Constitutes Function |
| A description/proof of that formula that uniquely maps each element of set into set constitutes function |
| 260: Order of Powers |
| A description/proof of order of powers |
| 261: For 2 Sets, Collection of Relations Between Sets Is Set |
| A description/proof of that for 2 sets, collection of relations between sets is set |
| 262: Finite Product of Sets Is Set |
| A description/proof of that finite product of sets is set |
| 263: For 2 Sets, Collection of Functions Between Sets Is Set |
| A description/proof of that for 2 sets, collection of functions between sets is set |
| 264: For Transfinite Recursion Theorem, Some Conditions with Which Partial Specifications of Formula Are Sufficient |
| A description/proof of for transfinite recursion theorem, some conditions with which partial specifications of formula are sufficient |
| 265: Map Between Topological Spaces Is Continuous at Point if They Are Subspaces of C^\infty Manifolds and There Are Charts of Manifolds Around Point and Point Image and Map Between Chart Open Subsets Which Is Restricted to Original Map Whose Restricted Coordinates Function Is Continuous |
| description/proof of that map between topological spaces is continuous at point if they are subspaces of \(C^\infty\) manifolds and there are charts of manifolds around point and point image and map between chart open subsets which is restricted to original map whose restricted coordinates function is continuous |
| 266: Inverse of Partial Ordering Is Partial Ordering |
| A description/proof of that inverse of partial ordering is partial ordering |
| 267: Minimal Element of Set w.r.t. Inverse of Ordering Is Maximal Element of Set w.r.t. Original Ordering |
| A description/proof of that minimal element of set w.r.t. inverse of ordering is maximal element of set w.r.t. original ordering |
| 268: Maximal Element of Set w.r.t. Inverse of Ordering Is Minimal Element of Set w.r.t. Original Ordering |
| A description/proof of that maximal element of set w.r.t. inverse of ordering is minimal element of set w.r.t. original ordering |
| 269: Inverse of Closed Bijection Is Continuous |
| A description/proof of that inverse of closed bijection is continuous |
| 270: Topological Space Is Compact Iff for Every Collection of Closed Subsets for Which Intersection of Any Finite Members Is Not Empty, Intersection of Collection Is Not Empty |
| A description/proof of that topological space is compact iff for every collection of closed subsets for which intersection of any finite members is not empty, intersection of collection is not empty |
| 271: Finite Product of Topological Spaces Equals Sequential Products of Topological Spaces |
| A description/proof of that finite product of topological spaces equals sequential products of topological spaces |
| 272: Finite Product of Compact Topological Spaces Is Compact |
| A description/proof of that finite product of compact topological spaces is compact |
| 273: For Transitive Set with Partial Ordering by Membership, Element Is Initial Segment Up to It |
| A description/proof of that for transitive set with partial ordering by membership, element is initial segment up to it |
| 274: Ordinal Number Is Grounded and Its Rank Is Itself |
| A description/proof of that ordinal number is grounded and its rank is itself |
| 275: Transitive Closure of Subset |
| A definition of transitive closure of subset |
| 276: Transitive Closure of Subset Is Transitive Set That Contains Subset |
| A description/proof of that transitive closure of subset is transitive set that contains subset |
| 277: Net to Product Topological Space Converges to Point iff Each Projection After Net Converges to Component of Point |
| A description/proof of that net to product topological space converges to point iff each projection after net converges to component of point |
| 278: Subset of Product Topological Space Is Closed iff It Is Intersection of Finite Unions of Products of Closed Subsets Only Finite of Which Are Not Whole Spaces |
| A description/proof of that subset of product topological space is closed iff it is intersection of finite unions of products of closed subsets only finite of which are not whole spaces |
| 279: Product of Connected Topological Spaces Is Connected |
| A description/proof of that product of connected topological spaces is connected |
| 280: No Set Has Itself as Member |
| A description/proof of that no set has itself as member |
| 281: No 2 Sets Have Each Other as Members |
| A description/proof of that no 2 sets have each other as members |
| 282: Product of Path-Connected Topological Spaces Is Path-Connected |
| A description/proof of that product of path-connected topological spaces is path-connected |
| 283: For Sequence on Topological Space, Around Point, There Is Open Set That Contains Only Finite Points of Sequence if No Subsequence Converges to Point |
| A description/proof of that for sequence on topological space, around point, there is open set that contains only finite points of sequence if no subsequence converges to point |
| 284: Relation Between Power Set Axiom and Subset Axiom |
| A description of relation between power set axiom and subset axiom |
| 285: Some Parts of Legitimate Formulas for ZFC Set Theory |
| A description/proof of some parts of legitimate formulas for ZFC set theory |
| 286: For Map from Topological Space to Metric Space, Image of Closed Set Is Closed on Image of Domain, if for Any Sequence on Closed Set for Which Image of Sequence Converges on Image of Domain, Convergent Point Is on Image of Closed Set |
| A description/proof of that for map from topological space into metric space, image of closed set is closed on image of domain, if for any sequence on closed set for which image of sequence converges on image of domain, convergent point is on image of closed set |
| 287: Unbounded Collection of Ordinal Numbers Is Not Set |
| A description/proof of that unbounded collection of ordinal numbers is not set |
| 288: For Injective Monotone Continuous Operation from Ordinal Numbers Collection into Ordinal Numbers Collection and Subset of Range, Union of Subset Is in Range |
| A description/proof of that for injective monotone continuous operation from ordinal numbers collection into ordinal numbers collection and image of subset of domain, union of image is in range |
| 289: Closed Set Minus Open Set Is Closed |
| A description/proof of that closed set minus open set is closed |
| 290: Compactness of Topological Subset as Subset Equals Compactness as Subspace |
| A description/proof of that compactness of topological subset as subset equals compactness as subspace |
| 291: Coordinates Matrix of Inverse Riemannian Metric Is Inverse of Coordinates Matrix of Riemannian Metric |
| A description/proof of that coordinates matrix of inverse Riemannian metric is inverse of coordinates matrix of Riemannian metric |
| 292: For Monotone Operation from Ordinal Numbers Collection into Ordinal Numbers Collection, Value Equals or Contains Argument |
| A description/proof of that for monotone operation from ordinal numbers collection into ordinal numbers collection, value equals or contains argument |
| 293: Fixed-Point in Proof of Veblen Fixed-Point Theorem Is Smallest That Satisfies Condition |
| A description/proof of that fixed-point in proof of Veblen fixed-point theorem is smallest that satisfies condition |
| 294: Derived Operation of Monotone Continuous Operation from Ordinal Numbers Collection into Ordinal Numbers Collection Is Monotone Continuous |
| A description/proof of that derived operation of monotone continuous operation from ordinal numbers collection into ordinal numbers collection is monotone continuous |
| 295: For Topological Space, Compact Subset of Subspace Is Compact on Base Space |
| A description/proof of that for topological space, compact subset of subspace is compact on base space |
| 296: For Topological Space, Subset of Compact Subset Is Not Necessarily Compact |
| A description/proof of that for topological space, subset of compact subset is not necessarily compact |
| 297: Quotient of Cylinder with Antipodal Points Identified Is Homeomorphic to Möbius Band |
| A description/proof of that quotient of cylinder with antipodal points identified is homeomorphic to Möbius Band |
| 298: For Topological Space, Intersection of Compact Subset and Subspace Is Not Necessarily Compact on Subspace |
| A description/proof of that for topological space, intersection of compact subset and subspace is not necessarily compact on subspace |
| 299: For Locally Compact Hausdorff Topological Space, Around Point, There Is Open Neighborhood Whose Closure Is Compact |
| A description/proof of that for locally compact Hausdorff topological space, around point, there is open neighborhood whose closure is compact |
| 300: Intersection of Closure of Subset and Open Subset Is Contained in Closure of Intersection of Subset and Open Subset |
| A description/proof of that intersection of closure of subset and open subset is contained in closure of intersection of subset and open subset |
| 301: For Topological Space and Point on Subspace, Intersection of Neighborhood of Point on Base Space and Subspace Is Neighborhood on Subspace |
| A description/proof of that for topological space and point on subspace, intersection of neighborhood of point on base space and subspace is neighborhood on subspace |
| 302: For Topological Space, Subspace Subset That Is Compact on Base Space Is Compact on Subspace |
| A description/proof of that for topological space, subspace subset that is compact on base space is compact on subspace |
| 303: Closed Subspace of Locally Compact Topological Space Is Locally Compact |
| A description/proof of that closed subspace of locally compact topological space is locally compact |
| 304: Open Subspace of Locally Compact Hausdorff Topological Space Is Locally Compact |
| A description/proof of that open subspace of locally compact Hausdorff topological space is locally compact |
| 305: Topological Subspace Is Locally Closed Iff It Is Intersection of Closed Subset and Open Subset of Base Space |
| A description/proof of that topological subspace is locally closed iff it is intersection of closed subset and open subset of base space |
| 306: 1 Point Subset of Hausdorff Topological Space Is Closed |
| A description/proof of that 1 point subset of Hausdorff topological space is closed |
| 307: Well-Ordered Set |
| A definition of well-ordered set |
| 308: Chain in Set |
| A definition of chain in set |
| 309: Partially-Ordered Set |
| A definition of partially-ordered set |
| 310: Linearly-Ordered Set |
| A definition of linearly-ordered set |
| 311: Maximal Element of Set |
| A definition of maximal element of set |
| 312: Well-Ordered Subset with Inclusion Ordering Is Chain in Base Set |
| A description/proof of that well-ordered subset with inclusion ordering is chain in base set |
| 313: Hausdorff Maximal Principle: Chain in Partially-Ordered Set Is Contained in Maximal Chain |
| A description/proof of that Hausdorff maximal principle: chain in partially-ordered set is contained in maximal chain |
| 314: Product of Topological Subspaces Is Subspace of Product of Base Spaces |
| A description/proof of that product of topological subspaces is subspace of product of base spaces |
| 315: For Locally Finite Open Cover of Topological Space, Closure of Union of Open Sets Is Union of Closures of Open Sets |
| A description/proof of that for locally finite open cover of topological space, closure of union of open sets is union of closures of open sets |
| 316: Closure of Union of Finite Subsets Is Union of Closures of Subsets |
| A description/proof of that closure of union of finite subsets is union of closures of subsets |
| 317: For Locally Finite Cover of Topological Space, Compact Subset Intersects Only Finite Elements of Cover |
| A description/proof of that for locally finite cover of topological space, compact subset intersects only finite elements of cover |
| 318: Topological Sum of Paracompact Topological Spaces Is Paracompact |
| A description/proof of that topological sum of paracompact topological spaces is paracompact |
| 319: For Monotone Continuous Operation from Ordinal Numbers Collection into Ordinal Numbers Collection, Image of Limit Ordinal Number Is Limit Ordinal Number |
| A description/proof of that for monotone continuous operation from ordinal numbers collection into ordinal numbers collection, image of limit ordinal number is limit ordinal number |
| 320: Ordinal Number Is Limit Ordinal Number iff It Is Nonzero and Is Union of Its All Members |
| A description/proof of that ordinal number is limit ordinal number iff it is nonzero and is union of its all members |
| 321: For Monotone Ordinal Numbers Operation, 2 Domain Elements Are in Membership Relation if Corresponding Images Are in Same Relation |
| A description/proof of that for monotone ordinal numbers operation, 2 domain elements are in membership relation if corresponding images are in same relation |
| 322: For Well-Ordered Structure and Its Sub Structure, Ordinal Number of Sub Structure Is Member of or Is Ordinal Number of Base Structure |
| A description/proof of that for well-ordered structure and its sub structure, ordinal number of sub structure is member of or is ordinal number of base structure |
| 323: Locally Compact Hausdorff Topological Space Is Paracompact iff Space Is Disjoint Union of Open \sigma-Compact Subspaces |
| A description/proof of that locally compact Hausdorff topological space is paracompact iff space is disjoint union of open \(\sigma\)-compact subspaces |
| 324: Descending Sequence of Ordinal Numbers Is FiniteDescending Sequence of Ordinal Numbers Is Finite |
| A description/proof of that descending sequence of ordinal numbers is finite |
| 325: Intersection of Set of Transitive Relations Is Transitive |
| A description/proof of that intersection of set of transitive relations is transitive |
| 326: Cantor Normal Form Is Unique |
| A description/proof of that Cantor normal form is unique |
| 327: Projective Hyperplane Is Hausdorff |
| A description/proof of that projective hyperplane is Hausdorff |
| 328: For Normal Topological Space, Collapsed Topological Space by Closed Subset Is Normal |
| A description/proof of that for normal topological space, collapsed topological space by closed subset is normal |
| 329: For Regular Topological Space, Collapsed Topological Space by Closed Subset Is Hausdorff |
| A description/proof of that for regular topological space, collapsed topological space by closed subset is Hausdorff |
| 330: Inclusion into Topological Space from Subspace Is Continuous |
| A description/proof of that inclusion into topological space from subspace is continuous |
| 331: Map Between Topological Spaces Is Continuous iff Preimage of Each Closed Subset of Codomain Is Closed |
| A description/proof of that map between topological spaces is continuous iff preimage of each closed subset of codomain is closed |
| 332: Inclusion into Topological Space from Closed Subspace Is Closed Continuous Embedding |
| A description/proof of that inclusion into topological space from closed subspace is closed continuous embedding |
| 333: Map from Mapping Cylinder into Topological Space Is Continuous iff Induced Maps from Adjunction Attaching Origin Space and from Adjunction Attaching Destination Space Are Continuous |
| A description/proof of that map from mapping cylinder into topological space is continuous iff induced maps from adjunction attaching origin space and from adjunction attaching destination space are continuous |
| 334: Closure of Subgroup of Topological Group Is Subgroup |
| A description/proof of that closure of subgroup of topological group is subgroup |
| 335: Linear Map Between Euclidean Topological Spaces Is Continuous |
| A description/proof of that linear map between Euclidean topological spaces is continuous |
| 336: Closure of Normal Subgroup of Topological Group Is Normal Subgroup |
| A description/proof of that closure of normal subgroup of topological group is normal subgroup |
| 337: For Coset Map with Respect to Subgroup, Preimage of Image of Subset Is Subgroup Multiplied by Subset |
| A description/proof of that for coset map with respect to subgroup, preimage of image of subset is subgroup multiplied by subset |
| 338: With Respect to Subgroup, Coset by Element of Group Equals Coset iff Element Is Member of Latter Coset |
| A description/proof of that with respect to subgroup, coset by element of group equals coset iff element is member of latter coset |
| 339: With Respect to Normal Subgroup, Set Of Cosets Forms Group |
| A description/proof of that with respect to normal subgroup, set of cosets forms group |
| 340: For Group, Symmetric Subset, Element of Group, and Subset, Element Multiplied by Symmetric Subset from Right or Left and Symmetric Subset Multiplied by Subset from Right or Left Are Disjoint if Element Multiplied by Symmetric Subset from Left and Right and Subset Are Disjoint |
| A description/proof of that for group, symmetric subset, element of group, and subset, element multiplied by symmetric subset from right or left and symmetric subset multiplied by subset from right or left are disjoint if element multiplied by symmetric subset from left and right and subset are disjoint |
| 341: Multiplication of Matrix Made of Same Size Blocks by Matrix Made of Multiplicable Same Size Blocks Is Blocks-Wise |
| A description/proof of that multiplication of matrix made of same size blocks by matrix made of multiplicable same size blocks is blocks-wise |
| 342: Set of n x n Quaternion Matrices Is 'Rings - Homomorphism Morphisms' Isomorphic to Set of Corresponding 2n x 2n Complex Matrices |
| A description/proof of that set of n x n quaternion matrices is 'rings - homomorphism morphisms' isomorphic to set of corresponding 2n x 2n complex matrices |
| 343: n-Dimensional Quaternion General Linear Group Is 'Groups - Homomorphism Morphisms' Isomorphic to Set of Nonzero Determinant Corresponding 2nx2n Complex Matrices and Can Be Represented by Latter |
| A description/proof of that n-dimensional quaternion general linear group is 'groups - homomorphism morphisms' isomorphic to set of nonzero determinant corresponding 2n x 2n complex matrices and can be represented by latter |
| 344: Topological Space Is Connected if Quotient Space and Each Element of Quotient Space Are Connected |
| A description/proof of that topological space is connected if quotient space and each element of quotient space are connected |
| 345: 2 x 2 Special Orthogonal Matrix Can Be Expressed with Sine and Cosine of Angle |
| A description/proof of that 2 x 2 special orthogonal matrix can be expressed with sine and cosine of angle |
| 346: 2 x 2 Special Unitary Matrix Can Be Expressed with Sine and Cosine of Angle and Imaginary Exponentials of 2 Angles |
| A description/proof of that 2 x 2 special unitary matrix can be expressed with sine and cosine of angle and imaginary exponentials of 2 angles |
| 347: Nonzero Multiplicative Translation from Complex Numbers Euclidean Topological Space onto Complex Numbers Euclidean Topological Space Is Homeomorphism |
| A description/proof of that nonzero multiplicative translation from complex numbers Euclidean topological space onto complex numbers Euclidean topological space is homeomorphism |
| 348: Conjugation from Complex Numbers Euclidean Topological Space onto Complex Numbers Euclidean Topological Space Is Homeomorphism |
| A description/proof of that conjugation from complex numbers Euclidean topological space onto complex numbers Euclidean topological space is homeomorphism |
| 349: Quotient Space of Compact Topological Space Is Compact |
| A description/proof of that quotient space of compact topological space is compact |
| 350: n-Sphere Is Path-Connected |
| A description/proof of that n-sphere is path-connected |
| 351: Finite Intersection of Open Dense Subsets of Topological Space Is Open Dense |
| A description/proof of that finite intersection of open dense subsets of topological space is open dense |
| 352: Minus Dedekind Cut Of Dedekind Cut Is Really Dedekind Cut |
| A description/proof of that minus Dedekind cut Of Dedekind cut is really Dedekind cut |
| 353: For Locally Compact Hausdorff Topological Space, in Neighborhood Around Point, There Is Open Neighborhood Whose Closure Is Compact and Contained in Neighborhood |
| A description/proof of that for locally compact Hausdorff topological space, in neighborhood around point, there is open neighborhood whose closure is compact and contained in neighborhood |
| 354: Complement of Empty-Interior Especially Nowhere Dense Subset Is Dense |
| A description/proof of that complement of empty-interior especially nowhere dense subset is dense |
| 355: Complement of Open Dense Subset Is Nowhere Dense |
| A description/proof of that complement of open dense subset is nowhere dense |
| 356: For Metric Space, Distance Between Points in 2 Open Balls Is Larger Than Distance Between Centers Minus Sum of Radii and Smaller Than Distance Between Centers Plus Sum of Radii |
| A description/proof of that for metric space, distance between points in 2 open balls is larger than distance between centers minus sum of radii and smaller than distance between centers plus sum of radii |
| 357: For Subset of Topological Space, Closure of Subset Minus Subset Has Empty Interior |
| A description/proof of that for subset of topological space, closure of subset minus subset has empty interior |
| 358: Subset of 1st Category Subset Is of 1st Category |
| A description/proof of that subset of 1st category subset is of 1st category |
| 359: Superset of Residual Subset Is Residual |
| A description/proof of that superset of residual subset is residual |
| 360: C^\infty Vectors Field on Regular Submanifold Is C^\infty as Vectors Field Along Regular Submanifold on Supermanifold |
| A description/proof of that \(C^\infty\) vectors field on regular submanifold is \(C^\infty\) as vectors field along regular submanifold on supermanifold |
| 361: Finite Union of Nowhere Dense Subsets of Topological Space Has Empty Interior |
| A description/proof of that finite union of nowhere dense subsets of topological space has empty interior |
| 362: There Are Rational and Irrational Dedekind Cuts Between 2 Dedekind Cuts |
| A description/proof of that there are rational and irrational Dedekind cuts between 2 Dedekind cuts |
| 363: For Product of 2 C^\infty Manifolds, Product for Which One of Constituents Is Replaced with Regular Submanifold Is Regular Submanifold |
| A description/proof of that for product of 2 \(C^\infty\) manifolds, product for which one of constituents is replaced with regular submanifold is regular submanifold |
| 364: Intersection of Products of Sets Is Product of Intersections of Sets |
| A description/proof of that intersection of products of sets is product of intersections of sets |
| 365: C^\infty Vectors Field Is Uniquely Defined by Its C^\infty Metric Value Functions with All C^\infty Vectors Fields |
| A description/proof of that \(C^\infty\) vectors field is uniquely defined by its \(C^\infty\) metric value functions with all \(C^\infty\) vectors fields |
| 366: For C^\infty Map Between C^\infty Manifolds, Restriction of Map on Regular Submanifold Domain and Regular Submanifold Codomian Is C^\infty |
| A description/proof of that for \(C^\infty\) map between \(C^\infty\) manifolds, restriction of map on regular submanifold domain and regular submanifold codomain Is \(C^\infty\) |
| 367: For C^\infty Manifold and Its Regular Submanifold, Open Subset of Super Manifold Is C^\infty Manifold and Intersection of Open Subset and Regular Submanifold Is Regular Submanifold of Open Subset Manifold |
| A description/proof of that for \(C^\infty\) manifold and its regular submanifold, open subset of super manifold is \(C^\infty\) manifold and intersection of open subset and regular submanifold is regular submanifold of open subset manifold |
| 368: Restriction of C^\infty Vectors Bundle on Regular Submanifold Base Space Is C^\infty Vectors Bundle |
| A description/proof of that restriction of \(C^\infty\) vectors bundle on regular submanifold base space is \(C^\infty\) vectors bundle |
| 369: C^\infty Function on C^\infty Manifold Is C^\infty on Regular Submanifold |
| A description/proof of that \(C^\infty\) function on \(C^\infty\) manifold is \(C^\infty\) on regular submanifold |
| 370: For Euclidean C^\infty Manifold and Its Regular Submanifold, Vectors Field Along Regular Submanifold Is C^\infty iff Its Components w.r.t. Standard Chart Are C^\infty on Regular Submanifold |
| A description/proof of that for Euclidean \(C^\infty\) manifold and its regular submanifold, vectors field along regular submanifold is \(C^\infty\) iff its components w.r.t. standard chart are \(C^\infty\) on regular submanifold |
| 371: Restriction of C^\infty Map on Open Domain and Open Codomain Is C^\infty |
| A description/proof of that restriction of \(C^\infty\) map on open domain and open codomain Is \(C^\infty\) |
| 372: Functionally Structured Topological Spaces Category Morphisms Are Morphisms |
| A description/proof of that functionally structured topological spaces category morphisms are morphisms |
| 373: Induced Functional Structure on Continuous Topological Spaces Map Codomain Is Functional Structure |
| A description/proof of that induced functional structure on continuous topological spaces map codomain is functional structure |
| 374: Induced Functional Structure on Topological Subspace by Inclusion Is Functional Structure |
| A description/proof of that induced functional structure on topological subspace by inclusion is functional structure |
| 375: For 1st Countable Topological Space, Some Facts About Points Sequences and Subset |
| A description/proof of that for 1st countable topological space, some facts about points sequences and subset |
| 376: Characteristic Property of Subspace Topology |
| A description/proof of characteristic property of subspace topology |
| 377: Characteristic Property of Product Topology |
| A description/proof of characteristic property of product topology |
| 378: Characteristic Property of Disjoint Union |
| A description/proof of characteristic property of disjoint union |
| 379: Continuous Surjection Between Topological Spaces Is Quotient Map if Any Codomain Subset Is Closed if Its Preimage Is Closed |
| A description/proof of that continuous surjection between topological spaces is quotient map if any codomain subset is closed if its preimage is closed |
| 380: For Quotient Map, Its Restriction on Open or Closed Saturated Domain and on Restricted Image Codomain Is Quotient Map |
| A description/proof of that for quotient map, its restriction on open or closed saturated domain and on restricted image codomain is quotient map |
| 381: Categories Equivalence Is Equivalence Relation |
| A description/proof of that categories equivalence is equivalence relation |
| 382: Preimage Under Surjection Is Saturated w.r.t. Surjection |
| A description/proof of that preimage under surjection is saturated w.r.t. surjection |
| 383: Dichotomically Disjoint Set of Sets |
| A definition of dichotomically disjoint set of sets |
| 384: For Set of Sets, Dichotomically Nondisjoint Does Not Necessarily Mean Pair-Wise Nondisjoint |
| A description/proof of that for set of sets, dichotomically nondisjoint does not necessarily mean pair-wise nondisjoint |
| 385: Union of Dichotomically Nondisjoint Set of Real Intervals Is Real Interval |
| A description/proof of that union of dichotomically nondisjoint set of real intervals is real interval |
| 386: Connected Topological Subspaces of 1-Dimensional Euclidean Topological Space Are Intervals |
| A description/proof of that connected topological subspaces of 1-dimensional Euclidean topological space are intervals |
| 387: For Topological Space, Open and Closed Subset of Space Is Union of Connected Components of Space |
| A description/proof of that for topological space, open and closed subset of space is union of connected components of space |
| 388: For Hausdorff Topological Space and 2 Disjoint Compact Subsets, There Are Disjoint Open Subsets Each of Which Contains Compact Subset |
| A description/proof of that for Hausdorff topological space and 2 disjoint compact subsets, there are disjoint open subsets each of which contains compact subset |
| 389: On 2nd-Countable Topological Space, Open Cover Has Countable Subcover |
| A description/proof of that on 2nd-countable topological space, open cover has countable subcover |
| 390: Topological Space Is Countably Compact iff Each Infinite Subset Has \omega-Accumulation Point |
| A description/proof of that topological space is countably compact iff each infinite subset has \(\omega\)-accumulation point |
| 391: Topological Space Is Countably Compact if It Is Sequentially Compact |
| A description/proof of that topological space is countably compact if it is sequentially compact |
| 392: 1st-Countable Topological Space Is Sequentially Compact if It Is Countably Compact |
| A description/proof of that 1st-countable topological space is sequentially compact if it is countably compact |
| 393: Continuous Map from Compact Topological Space into Hausdorff Topological Space Is Proper |
| A description/proof of that continuous map from compact topological space into Hausdorff topological space is proper |
| 394: Composition of Preimage After Map of Subset Contains Argument Set |
| A description/proof of that composition of preimage after map of subset contains argument set |
| 395: Closed Continuous Map Between Topological Spaces with Compact Fibers Is Proper |
| A description/proof of that closed continuous map between topological spaces with compact fibers is proper |
| 396: Continuous Embedding Between Topological Spaces with Closed Range Is Proper |
| A description/proof of that continuous embedding between topological spaces with closed range is proper |
| 397: Continuous Map from Topological Space into Hausdorff Topological Space with Continuous Left Inverse Is Proper |
| A description/proof of that continuous map from topological space into Hausdorff topological space with continuous left inverse is proper |
| 398: Restriction of Proper Map Between Topological Spaces on Saturated Domain Subset and Range Codomain Is Proper |
| A description/proof of that restriction of proper map between topological spaces on saturated domain subset and range codomain is proper |
| 399: For 'Independent Variable'-Value Pairs Data, Choosing Origin-Passing Approximating Line with Least Value Difference Squares Sum Equals Projecting Values Vector to Independent Variables Vector Line |
| A description/proof of that for 'independent variable'-value pairs data, choosing origin-passing approximating line with least value difference squares sum equals projecting values vector to independent variables vector line |
| 400: For Complete Metric Space, Closed Subspace Is Complete |
| A description/proof of that for complete metric space, closed subspace is complete |
| 401: On T_1 Topological Space, Point Is \omega-Accumulation Point of Subset iff It Is Accumulation Point of Subset |
| A description/proof of that on \(T_1\) topological space, point is \(\omega\)-accumulation point of subset iff it is accumulation point of subset |
| 402: Metric Space Is Compact iff Each Infinite Subset Has \omega-Accumulation Point |
| A description/proof of that metric space is compact iff each infinite subset has \(\omega\)-accumulation point |
| 403: For C^\infty Vectors Bundle, Global Connection Can Be Constructed with Local Connections over Open Cover, Using Partition of Unity Subordinate to Open Cover |
| A description/proof of that for \(C^\infty\) vectors bundle, global connection can be constructed with local connections over open cover, using partition of unity subordinate to open cover |
| 404: Riemannian Bundle Has Compatible Connection |
| A description/proof of that Riemannian bundle has compatible connection |
| 405: Map from Open Subset of C^\infty Manifold onto Open Subset of Euclidean Topological Space Is Chart Map iff It Is Diffeomorphism |
| A description/proof of that map from open subset of \(C^\infty\) manifold onto open subset of Euclidean \(C^\infty\) manifold is chart map iff it is diffeomorphism |
| 406: %Field Name% Vectors Space |
| A definition of %field name% vectors space |
| 407: For Vectors Bundle, Trivializing Open Subset Is Not Necessarily Chart Open Subset, but There Is Possibly Smaller Chart Trivializing Open Subset at Any Point on Trivializing Open Subset |
| description/proof of that for \(C^\infty\) vectors bundle, trivializing open subset is not necessarily chart open subset, but there is possibly smaller chart trivializing open subset at each point on trivializing open subset |
| 408: For Vectors Bundle, There Is Chart Trivializing Open Cover |
| description/proof of that for \(C^\infty\) vectors bundle, there is chart trivializing open cover |
| 409: For \(C^\infty\) Vectors Bundle, Trivialization of Chart Trivializing Open Subset Induces Canonical Chart Map |
| description/proof of that for \(C^\infty\) vectors bundle, trivialization of chart trivializing open subset induces canonical chart map |
| 410: For Vectors Bundle, Section over Trivializing Open Subset Is C^\infty iff Coefficients w.r.t. C^\infty Frame over There Are C^infty |
| description/proof of that for \(C^\infty\) vectors bundle, section over trivializing open subset is \(C^\infty\) iff coefficients w.r.t. \(C^\infty\) frame over there are \(C^\infty\) |
| 411: %Structure Kind Name% Endomorphism |
| A definition of %structure kind name% endomorphism |
| 412: For Vectors Bundle, Chart Open Subset on Base Space Is Not Necessarily Trivializing Open Subset (Probably) |
| description/proof of that for \(C^\infty\) vectors bundle, chart open subset on base space is not necessarily trivializing open subset (probably) |
| 413: For Vectors Bundle, C^\infty Frame Exists Over and Only Over Trivializing Open Subset |
| description/proof of that for \(C^\infty\) vectors bundle, \(C^\infty\) frame exists over and only over trivializing open subset |
| 414: Compositions of Homotopic Maps Are Homotopic |
| A description/proof of that compositions of homotopic maps are homotopic |
| 415: Fundamental Group Homomorphism Induced by Composition of Continuous Maps Is Composition of Fundamental Group Homomorphisms Induced by Maps |
| A description/proof of that fundamental group homomorphism induced by composition of continuous maps is composition of fundamental group homomorphisms induced by maps |
| 416: Fundamental Group Homomorphism Induced by Homeomorphism Is 'Groups - Group Homomorphisms' Isomorphism |
| A description/proof of that fundamental group homomorphism induced by homeomorphism is 'groups - group homomorphisms' isomorphism |
| 417: Fundamental Theorem for Group Homomorphism |
| A description/proof of fundamental theorem for group homomorphism |
| 418: Canonical Map from Fundamental Group on Finite Product Topological Space into Product of Constituent Topological Space Fundamental Groups Is 'Groups - Group Homomorphisms' Isomorphism |
| A description/proof of that canonical map from fundamental group on finite product topological space into product of constituent topological space fundamental groups is 'groups - group homomorphisms' isomorphism |
| 419: Fundamental Group Homomorphism Induced by Homotopy Equivalence Is 'Groups - Group Homomorphisms' Isomorphism |
| A description/proof of that fundamental group homomorphism induced by homotopy equivalence is 'groups - group homomorphisms' isomorphism |
| 420: 2 Continuous Maps from Connected Topological Space Such That, for Any Point, if They Agree at Point, They Agree on Neighborhood and if They Disagree at Point, They Disagree on Neighborhood, Totally Agree or Totally Disagree |
| A description/proof of that 2 continuous maps from connected topological space such that, for any point, if they agree at point, they agree on neighborhood and if they disagree at point, they disagree on neighborhood, totally agree or totally disagree |
| 421: For 2 Path-Connected Points on Topological Space, There Is 'Groups - Group Homomorphisms' Isomorphism Between Fundamental Groups That Multiplies Inverse-Path Class from Left and Path Class from Right in Path Classes Groupoid |
| A description/proof of that for 2 path-connected points on topological space, there is 'groups - group homomorphisms' isomorphism between fundamental groups that multiplies inverse-path class from left and path class from right in path classes groupoid |
| 422: For 2 Homotopic Maps, Point on Domain, and Fundamental Group Homomorphisms Induced by Maps, 2nd Homomorphism Is Composition of Canonical 'Groups - Group Homomorphisms' Isomorphism Between Codomains of Homomorphisms After 1st Homomorphism |
| A description/proof of that for 2 homotopic maps, point on domain, and fundamental group homomorphisms induced by maps, 2nd homomorphism is composition of canonical 'groups - group homomorphisms' isomorphism between codomains of homomorphisms after 1st homomorphism |
| 423: For Finite-Product Topological Space, Product of Neighborhoods Is Neighborhood |
| A description/proof of that for finite-product topological space, product of neighborhoods is neighborhood |
| 424: Finite Product of Locally Compact Topological Spaces Is Locally Compact |
| A description/proof of that finite product of locally compact topological spaces is locally compact |
| 425: For Product Topological Space, Projection of Compact Subset Is Compact |
| A description/proof of that for product topological space, projection of compact subset is compact |
| 426: Euclidean Vectors Space |
| A definition of Euclidean vectors space |
| 427: Euclidean Norm on Euclidean Vectors Space |
| definition of Euclidean norm on Euclidean vectors space |
| 428: Euclidean-Normed Euclidean Vectors Space |
| A definition of Euclidean-normed Euclidean vectors space |
| 429: 2 Continuous Maps into Hausdorff Topological Space That Disagree at Point Disagree on Neighborhood of Point |
| A description/proof of that 2 continuous maps into Hausdorff topological space that disagree at point disagree on neighborhood of point |
| 430: 2 Continuous Maps from Connected Topological Space into Hausdorff Topological Space Such That, for Any Point, if They Agree at Point, They Agree on Neighborhood, Totally Agree or Totally Disagree |
| A description/proof of that 2 continuous maps from connected topological space into Hausdorff topological space such that, for any point, if they agree at point, they agree on neighborhood, totally agree or totally disagree |
| 431: Complete Metric Space |
| A definition of complete metric space |
| 432: Interior of Subset of Topological Space |
| A definition of interior of subset of topological space |
| 433: Bijection |
| A definition of bijection |
| 434: Union of Indexed Subsets Minus Union of Subsets Indexed with Same Indices Set Is Contained in Union of Subset Minus Subset for Each Index |
| A description/proof of that union of indexed subsets minus union of subsets indexed with same indices set is contained in union of subset minus subset for each index |
| 435: Subset Minus Subset Is Complement of 2nd Subset Minus Complement of 1st Subset |
| A description/proof of that subset minus subset is complement of 2nd subset minus complement of 1st subset |
| 436: Product Set |
| A definition of product set |
| 437: Convergence of Sequence on Metric Space |
| A definition of convergence of sequence on metric space |
| 438: Cauchy Sequence on Metric Space |
| A definition of Cauchy sequence on metric space |
| 439: Continuous Image of Path-Connected Subspace of Domain Is Path-Connected on Codomain |
| A description/proof of that continuous image of path-connected subspace of domain is path-connected on codomain |
| 440: Rotation in n-Dimensional Euclidean Vectors Space Is Same 2-Dimensional Rotations Along (n - 2)-Dimensional Subspace Axis |
| A description/proof of that rotation in \(n\)-dimensional Euclidean vectors space is same \(2\)-dimensional rotations along \((n - 2)\)-dimensional subspace axis |
| 441: Dense Subset of Topological Space |
| A definition of dense subset of topological space |
| 442: Nowhere Dense Subset of Topological Space |
| A definition of nowhere dense subset of topological space |
| 443: Separable Topological Space |
| A definition of separable topological space |
| 444: Banach Space |
| A definition of Banach space |
| 445: For Covering Map, 2 Lifts of Continuous Map from Connected Topological Space Totally Agree or Totally Disagree |
| A description/proof of that for covering map, 2 lifts of continuous map from connected topological space totally agree or totally disagree |
| 446: Metric Induced by Norm on Real or Complex Vectors Space |
| A definition of metric induced by norm on real or complex vectors space |
| 447: Normal Subgroup of Group |
| A definition of normal subgroup of group |
| 448: Polish Space |
| A definition of Polish space |
| 449: Vectors Field on Restricted Tangent Vectors Bundle Is C^\infty iff Operation Result on Any C^\infty Function on Super Manifold Is C^\infty on Regular Submanifold |
| A description/proof of that vectors field on restricted tangent vectors bundle is \(C^\infty\) iff operation result on any \(C^\infty\) function on super manifold is \(C^\infty\) on regular submanifold |
| 450: For Covering Map, There Is Unique Lift of Continuous Map from Finite Product of Closed Real Intervals for Each Initial Value |
| A description/proof of that for covering map, there is unique lift of continuous map from finite product of closed real intervals for each initial value |
| 451: Chart on Topological Manifold |
| A definition of chart on topological manifold |
| 452: Chart on \(C^\infty\) Manifold |
| definition of chart on \(C^\infty\) manifold |
| 453: For Map C^\infty at Point, Coordinates Function with Any Charts Is C^\infty at Point Image |
| A description/proof of that for map \(C^\infty\) at point, coordinates function with any charts is \(C^\infty\) at point image |
| 454: For Covering Map, There Is Unique Lift of Path for Each Point in Covering Map Preimage of Path Image of Point on Path Domain |
| A description/proof of that for covering map, there is unique lift of path for each point in covering map preimage of path image of point on path domain |
| 455: Lifts, That Start at Same Point, of Path-Homotopic Paths Are Path-Homotopic |
| A description/proof of that lifts, that start at same point, of path-homotopic paths are path-homotopic |
| 456: Maximal Atlas for Topological Manifold |
| A definition of maximal atlas for topological manifold |
| 457: Euclidean \(C^\infty\) Manifold |
| definition of Euclidean \(C^\infty\) manifold |
| 458: For Covering Map, Lift of Reverse of Path Is Reverse of Lift of Path |
| A description/proof of that for covering map, lift of reverse of path is reverse of lift of path |
| 459: For Covering Map, Lift of Product of Paths Is Product of Lifts of Paths |
| A description/proof of that for covering map, lift of product of paths is product of lifts of paths |
| 460: For Covering Map, Criterion for Lift of Continuous Map from Path-Connected Locally Path-Connected Topological Space to Exist |
| A description/proof of for covering map, criterion for lift of continuous map from path-connected locally path-connected topological space to exist |
| 461: Matrix Norm Induced by Vector Norms |
| A definition of matrix norm induced by vector norms |
| 462: Frobenius Matrix Norm |
| A definition of Frobenius matrix norm |
| 463: 2 Points on Different Connected Components Are Not Path-Connected |
| A description/proof of that 2 points on different connected components are not path-connected |
| 464: Vectors Field Along C^\infty Curve Is C^\infty iff Operation Result on Any C^\infty Function is C^\infty |
| A description/proof of that vectors field along \(C^\infty\) curve is \(C^\infty\) iff operation result on any \(C^\infty\) function is \(C^\infty\) |
| 465: Velocity Vectors Field Along C^\infty Curve Is C^\infty |
| A description/proof of that velocity vectors field along \(C^\infty\) curve is \(C^\infty\) |
| 466: Covering Map |
| A definition of covering map |
| 467: Map from Open Subset of Euclidean C^\infty Manifold into Subset of Euclidean C^\infty Manifold C^k at Point, Where k Excludes 0 and Includes \infty |
| definition of map from open subset of Euclidean \(C^\infty\) manifold into subset of Euclidean \(C^\infty\) manifold \(C^k\) at point, where \(k\) excludes \(0\) and includes \(\infty\) |
| 468: Map Between Arbitrary Subsets of Euclidean C^\infty Manifolds C^k at Point, Where k Excludes 0 and Includes \infty |
| A definition of map between arbitrary subsets of Euclidean \(C^\infty\) manifolds \(C^k\) at point, where \(k\) excludes \(0\) and includes \(\infty\) |
| 469: Map Between Arbitrary Subsets of C^\infty Manifolds with Boundary C^k at Point, Where k Excludes 0 and Includes \infty |
| A definition of map between arbitrary subsets of \(C^\infty\) manifolds with boundary \(C^k\) at point, where \(k\) excludes \(0\) and includes \(\infty\) |
| 470: C^k Map Between Arbitrary Subsets of C^\infty Manifolds with Boundary , Where k Includes \infty |
| A definition of \(C^k\) map between arbitrary subsets of \(C^\infty\) manifolds with boundary, where \(k\) includes \(\infty\) |
| 471: For Maps Between Arbitrary Subsets of Euclidean C^\infty Manifolds C^k at Corresponding Points Composition Is C^k at Point |
| A description/proof of that for maps between arbitrary subsets of Euclidean \(C^\infty\) manifolds \(C^k\) at corresponding points, composition is \(C^k\) at point |
| 472: For Map Between Arbitrary Subsets of Euclidean C^\infty Manifolds C^k at Point, Restriction on Domain That Contains Point Is C^k at Point |
| A description/proof of that for map between arbitrary subsets of Euclidean \(C^\infty\) manifolds \(C^k\) at point, restriction on domain that contains point is \(C^k\) at point |
| 473: For Map Between Arbitrary Subsets of Euclidean C^infty Manifolds, Map Is C^k at Point if Restriction on Subspace Open Neighborhood of Point Domain Is C^k at Point |
| A description/proof of that for map between arbitrary subsets of Euclidean \(C^\infty\) manifolds, map is \(C^k\) at point if restriction on subspace open neighborhood of point domain is \(C^k\) at point |
| 474: For Map Between Arbitrary Subsets of C^\infty Manifolds with Boundary C^k at Point, Any Possible Pair of Domain Chart and Codomain Chart Satisfies Condition of Definition |
| A description/proof of that for map between arbitrary subsets of \(C^\infty\) manifolds with boundary \(C^k\) at point, any possible pair of domain chart and codomain chart satisfies condition of definition |
| 475: For Maps Between Arbitrary Subsets of \(C^\infty\) Manifolds with Boundary \(C^k\) at Corresponding Points, Composition Is \(C^k\) at Point |
| description/proof of that for maps between arbitrary subsets of \(C^\infty\) manifolds with boundary \(C^k\) at corresponding points, composition is \(C^k\) at point |
| 476: For Map Between Arbitrary Subsets of C^\infty Manifolds with Boundary C^k at Point, Restriction on Domain That Contains Point Is C^k at Point |
| A description/proof of that for map between arbitrary subsets of \(C^\infty\) manifolds with boundary \(C^k\) at point, restriction on domain that contains point is \(C^k\) at point |
| 477: For Map Between Arbitrary Subsets of C^\infty Manifolds with Boundary, Map Is C^k at Point if Restriction on Subspace Open Neighborhood of Point Domain Is C^k at Point |
| A description/proof of that for map between arbitrary subsets of \(C^\infty\) manifolds with boundary, map is \(C^k\) at point if restriction on subspace open neighborhood of point domain is \(C^k\) at point |
| 478: C^k-ness of Map from Closed Interval into Subset of Euclidean C-\infty Manifold at Boundary Point Equals Existence of One-Sided Derivatives with Continuousness, and Derivatives Are One-Sided Derivatives |
| A description/proof of that \(C^k\)-ness of map from closed interval into subset of Euclidean \(C^\infty\) manifold at boundary point equals existence of one-sided derivatives with continuousness, and derivatives are one-sided derivatives |
| 479: What Velocity of Curve at Closed Boundary Point Is |
| A description of what velocity of curve at closed boundary point is |
| 480: What Chart Induced Basis Vector on C^\infty Manifold with Boundary Is |
| A description of what chart induced basis vector on \(C^\infty\) manifold with boundary is |
| 481: Locally Topologically Closed Upper Half Euclidean Topological Space |
| A definition of locally topologically closed upper half Euclidean topological space |
| 482: Topological Manifold with Boundary |
| A definition of topological manifold with boundary |
| 483: Chart on Topological Manifold with Boundary |
| A definition of chart on topological manifold with boundary |
| 484: Maximal Atlas for Topological Manifold with Boundary |
| A definition of maximal atlas for topological manifold with boundary |
| 485: C^\infty Manifold with Boundary |
| A definition of \(C^\infty\) manifold with boundary |
| 486: Diffeomorphism Between Arbitrary Subsets of C^\infty Manifolds with Boundary |
| A definition of diffeomorphism between arbitrary subsets of \(C^\infty\) manifolds with boundary |
| 487: Compact Subset of Topological Space |
| A definition of compact subset of topological space |
| 488: Compact Topological Space |
| definition of compact topological space |
| 489: Differential of C^\infty Map Between C^\infty Manifolds with Boundary at Point |
| A definition of differential of \(C^\infty\) map between \(C^\infty\) manifolds with boundary at point |
| 490: For Diffeomorphism from C^\infty Manifold with Boundary onto Neighborhood of Point Image on C^\infty Manifold with Boundary, Differential at Point Is 'Vectors Spaces - Linear Morphisms' Isomorphism |
| A description/proof of that for diffeomorphism from \(C^\infty\) manifold with boundary onto neighborhood of point image on \(C^\infty\) manifold with boundary, differential at point is 'vectors spaces - linear morphisms' isomorphism |
| 491: Map Between Arbitrary Subsets of C^\infty Manifolds with Boundary Locally Diffeomorphic at Point |
| A definition of map between arbitrary subsets of \(C^\infty\) manifolds with boundary locally diffeomorphic at point |
| 492: Map Between Arbitrary Subsets of C^\infty Manifolds with Boundary Locally Diffeomorphic at Point Is C^\infty at Point |
| A description/proof of that map between arbitrary subsets of \(C^\infty\) manifolds with boundary locally diffeomorphic at point is \(C^\infty\) at point |
| 493: Map Between Arbitrary Subsets of C^\infty Manifolds with Boundary Bijective and Locally Diffeomorphic at Each Point Is Diffeomorphism |
| A description/proof of that map between arbitrary subsets of \(C^\infty\) manifolds with boundary bijective and locally diffeomorphic at each point is diffeomorphism |
| 494: Injective Map Between Topological Spaces Is Continuous Embedding if Domain Restriction of Map on Each Element of Open Cover Is Continuous Embedding onto Open Subset of Range or Codomain |
| A description/proof of that injective map between topological spaces is continuous embedding if domain restriction of map on each element of open cover is continuous embedding onto open subset of range or codomain |
| 495: For Intersection of 2 Subsets of Topological Space, Its Regarded as Subspace of a Subset as Subspace, Its Regarded as Subspace of Other Subset as Subspace, and Its Regarded as Subspace of Basespace Are Same |
| A description/proof of that for intersection of 2 subsets of topological space, its regarded as subspace of a subset as subspace, its regarded as subspace of other subset as subspace, and its regarded as subspace of basespace are same |
| 496: For Map Between Arbitrary Subsets of C^\infty Manifolds with Boundary C^k at Point, Restriction or Expansion on Codomain That Contains Range Is C^k at Point |
| A description/proof of that for map between arbitrary subsets of \(C^\infty\) manifolds with boundary \(C^k\) at point, restriction or expansion on codomain that contains range is \(C^k\) at point |
| 497: For Map Between Arbitrary Subsets of C^\infty Manifolds with Boundary Locally Diffeomorphic at Point, Restriction on Open Subset of Domain That Contains Point Is Locally Diffeomorphic at Point |
| A description/proof of that for map between arbitrary subsets of \(C^\infty\) manifolds with boundary locally diffeomorphic at point, restriction on open subset of domain that contains point is locally diffeomorphic at point |
| 498: For Maps Between Arbitrary Subsets of C^\infty Manifolds with Boundary Locally Diffeomorphic at Corresponding Points, Where Codomain of 1st Map Is Open Subset of Domain of 2nd Map, Composition Is Locally Diffeomorphic at Point |
| A description/proof of that for maps between arbitrary subsets of \(C^\infty\) manifolds with boundary locally diffeomorphic at corresponding points, where codomain of 1st map is open subset of domain of 2nd map, composition is locally diffeomorphic at point |
| 499: Ordered Pair |
| A definition of ordered pair |
| 500: Relation |
| A definition of relation |
| 501: Function |
| A definition of function |
| 502: Function over \(C^\infty\) Manifold with Boundary |
| definition of function over \(C^\infty\) manifold with boundary |
| 503: Tangent Vectors Space at Point |
| definition of tangent vectors space at point |
| 504: Subset Minus Union of Sequence of Subsets Is Intersection of Subsets Each of Which Is 1st Subset Minus Partial Union of Sequence |
| A description/proof of that subset minus union of sequence of subsets is intersection of subsets each of which is 1st subset minus partial union of sequence |
| 505: Pushforward Image of C^\infty Vectors Field Along Curve on Regular Submanifold into Supermanifold Under Inclusion Is C^\infty |
| A description/proof of that pushforward image of \(C^\infty\) vectors field along curve on regular submanifold into supermanifold under inclusion is \(C^\infty\) |
| 506: Rules of Structured Descriptions |
| The description of rules of structured descriptions |
| 507: Sequence |
| definition of sequence |
| 508: Permutation of Sequence |
| definition of permutation of sequence |
| 509: For Set of Sequences for Fixed Domain and Codomain, Permutation Bijectively Maps Set onto Set |
| description/proof of that for set of sequences for fixed domain and codomain, permutation bijectively maps set onto set |
| 510: Square of Euclidean Norm of \mathbb{R}^n Vector Is Equal to or Larger Than Positive Definite Real Quadratic Form Divided by Largest Eigenvalue |
| description/proof of that square of Euclidean norm of \(\mathbb{R}^n\) vector is equal to or larger than positive definite real quadratic form divided by largest eigenvalue |
| 511: Square of Euclidean Norm of \mathbb{R}^n Vector Is Equal to or Smaller Than Positive Definite Real Quadratic Form Divided by Smallest Eigenvalue |
| description/proof of that square of Euclidean norm of \(\mathbb{R}^n\) vector is equal to or smaller than positive definite real quadratic form divided by smallest eigenvalue |
| 512: Norm Induced by Inner Product on Real or Complex Vectors Space |
| definition of norm induced by inner product on real or complex vectors space |
| 513: Topology Induced by Metric |
| definition of topology induced by metric |
| 514: Euclidean Inner Product on Euclidean Vectors Space |
| definition of Euclidean inner product on Euclidean vectors space |
| 515: Euclidean Metric |
| definition of Euclidean metric |
| 516: Topological Subspace |
| definition of topological subspace |
| 517: Linear Map |
| definition of linear map |
| 518: Simply Connected Topological Space |
| definition of simply connected topological space |
| 519: Lift of Continuous Map by Covering Map |
| definition of lift of continuous map by covering map |
| 520: Covering Map into Simply Connected Topological Space Is Homeomorphism |
| description/proof of that covering map into simply connected topological space is homeomorphism |
| 521: Maps Homotopic Relative to Subset of Domain |
| definition of maps homotopic relative to subset of domain |
| 522: Homotopic Maps |
| definition of homotopic maps |
| 523: Contractible Topological Space |
| definition of contractible topological space |
| 524: Product Map |
| definition of product map |
| 525: Product Topological Space |
| definition of product topological space |
| 526: Map Preimage of Range Is Whole Domain |
| description/proof of that map preimage of range is whole domain |
| 527: For Infinite Product Topological Space and Subset, Point on Product Space Whose Each Finite-Components-Projection Belongs to Corresponding Projection of Subset Does Not Necessarily Belong to Subset |
| description/proof of that for infinite product topological space and subset, point on product space whose each finite-components-projection belongs to corresponding projection of subset does not necessarily belong to subset |
| 528: For Infinite Product Topological Space and Closed Subset, Point on Product Space Whose Each Finite-Components-Projection Belongs to Corresponding Projection of Subset Belongs to Subset |
| description/proof of that for infinite product topological space and closed subset, point on product space whose each finite-components-projection belongs to corresponding projection of subset belongs to subset |
| 529: Restriction of Continuous Embedding on Domain and Codomain Is Continuous Embedding |
| description/proof of that restriction of continuous embedding on domain and codomain is continuous embedding |
| 530: Sufficient Conditions for Existence of Unique Global Solution on Interval for Euclidean-Normed Euclidean Vectors Space ODE |
| description/proof of sufficient conditions for existence of unique global solution on interval for Euclidean-normed Euclidean vectors space ODE |
| 531: %Ring Name% Module |
| definition of %ring name% module |
| 532: Linearly Independent Subset of Module |
| definition of linearly independent subset of module |
| 533: For Finite Set of Points on Real Vectors Space, if for Point, Set of Subtractions of Point from Other Points Is Linearly Independent, It Is So for Each Point |
| description/proof of that for finite set of points on real vectors space, if for point, set of subtractions of point from other points are linearly independent, it is so for each point |
| 534: Affine-Independent Set of Points on Real Vectors Space |
| definition of affine-independent set of points on real vectors space |
| 535: Affine Combination of Possibly-Non-Affine-Independent Set of Base Points on Real Vectors Space |
| definition of affine combination of possibly-non-affine-independent set of base points on real vectors space |
| 536: Convex Combination of Possibly-Non-Affine-Independent Set of Base Points on Real Vectors Space |
| definition of convex combination of possibly-non-affine-independent set of base points on real vectors space |
| 537: Affine Set Spanned by Possibly-Non-Affine-Independent Set of Base Points on Real Vectors Space |
| definition of affine set spanned by possibly-non-affine-independent set of base points on real vectors space |
| 538: Convex Set Spanned by Possibly-Non-Affine-Independent Set of Base Points on Real Vectors Space |
| definition of convex set spanned by possibly-non-affine-independent set of base points on real vectors space |
| 539: Determinant of Square Matrix Whose Last Row Is All 1 and Whose Each Other Row Is All 0 Except Row Number + 1 Column 1 Is -1 to Power of Dimension + 1 |
| description/proof of that determinant of square matrix whose last row is all 1 and whose each other row is all 0 except row number + 1 column 1 is -1 to power of dimension + 1 |
| 540: Affine Set Spanned by Non-Affine-Independent Set of Base Points on Real Vectors Space Is Affine Set Spanned by Affine-Independent Subset of Base Points |
| description/proof of that affine set spanned by non-affine-independent set of base points on real vectors space is affine set spanned by affine-independent subset of base points |
| 541: Affine Simplex |
| definition of affine simplex |
| 542: Convex Set Spanned by Non-Affine-Independent Set of Base Points on Real Vectors Space Is Not Necessarily Affine Simplex Spanned by Affine-Independent Subset of Base Points |
| description/proof of that convex set spanned by non-affine-independent set of base points on real vectors space is not necessarily affine simplex spanned by affine-independent subset of base points |
| 543: Convex Set Spanned by Possibly-Non-Affine-Independent Set of Base Points on Real Vectors Space Is Convex |
| description/proof of that convex set spanned by possibly-non-affine-independent set of base points on real vectors space is convex |
| 544: Standard Simplex |
| definition of standard simplex |
| 545: Orientated Affine Simplex |
| definition of orientated affine simplex |
| 546: Face of Orientated Affine Simplex |
| definition of face of orientated affine simplex |
| 547: Affine Map from Affine Set Spanned by Possibly-Non-Affine-Independent Set of Base Points on Real Vectors Space |
| definition of affine map from affine set spanned by possibly-non-affine-independent set of base points on real vectors space |
| 548: Affine Map from Convex Set Spanned by Possibly-Non-Affine-Independent Set of Base Points on Real Vectors Space |
| definition of affine map from convex set spanned by possibly-non-affine-independent set of base points on real vectors space |
| 549: Affine Map from Affine or Convex Set Spanned by Possibly-Non-Affine-Independent Set of Base Points on Real Vectors Space Is Linear |
| description/proof of that affine map from affine or convex set spanned by possibly-non-affine-independent set of base points on real vectors space is linear |
| 550: Affine Subset of Real Vectors Space |
| definition of affine subset of real vectors space |
| 551: Convex Subset of Real Vectors Space |
| definition of convex subset of real vectors space |
| 552: Affine Subset of Finite-Dimensional Real Vectors Space Is Spanned by Finite Affine-Independent Set of Base Points |
| description/proof of that affine subset of finite-dimensional real vectors space is spanned by finite affine-independent set of base points |
| 553: When Convex Set Spanned by Non-Affine-Independent Set of Base Points on Real Vectors Space Is Affine Simplex, It Is Spanned by Affine-Independent Subset of Base Points |
| description/proof of that when convex set spanned by non-affine-independent set of base points on real vectors space is affine simplex, it is spanned by affine-independent subset of base points |
| 554: Face of Affine Simplex |
| definition of face of affine simplex |
| 555: Simplicial Complex |
| definition of simplicial complex |
| 556: Simplex Interior of Affine Simplex |
| definition of simplex interior of affine simplex |
| 557: Simplex Boundary of Affine Simplex |
| definition of simplex boundary of affine simplex |
| 558: Boundary of Subset of Topological Space |
| definition of boundary of subset of topological space |
| 559: Maximal Simplex in Simplicial Complex |
| definition of maximal simplex in simplicial complex |
| 560: Canonical Topology for Finite-Dimensional Real Vectors Space |
| definition of canonical topology for finite-dimensional real vectors space |
| 561: Canonical C^\infty Atlas for Finite-Dimensional Real Vectors Space |
| definition of canonical \(C^\infty\) atlas for finite-dimensional real vectors space |
| 562: Composition of Affine Maps Is Affine Map |
| description/proof of that composition of affine maps is affine map |
| 563: Ascending Sequence of Faces of Affine Simplex |
| definition of ascending sequence of faces of affine simplex |
| 564: For Linearly Independent Finite Subset of Module, Induced Subset of Module with Some Linear Combinations Is Linearly Independent |
| description/proof of that for linearly independent finite subset of module, induced subset of module with some linear combinations is linearly independent |
| 565: Subset of Affine-Independent Set of Points on Real Vectors Space Is Affine-Independent |
| description/proof of that subset of affine-independent set of points on real vectors space is affine-independent |
| 566: Barycenter of Affine Simplex |
| definition of barycenter of affine simplex |
| 567: For Affine Simplex and Ascending Sequence of Faces, Set of Barycenters of Faces Is Affine-Independent |
| description/proof of that for affine simplex and ascending sequence of faces, set of barycenters of faces is affine-independent |
| 568: For Simplicial Complex on Finite-Dimensional Real Vectors Space, Each Simplex in Complex Is Faces of Elements of Subset of Maximal Simplexes Set |
| description/proof of that for simplicial complex on finite-dimensional real vectors space, each simplex in complex is faces of elements of subset of maximal simplexes set |
| 569: For Affine Simplex, Ascending Sequence of Faces, and Set of Barycenters of Faces, Convex Combination of Subset of Set of Barycenters Is Convex Combination W.r.t. Set of Vertexes of Affine Simplex |
| description/proof of that for affine simplex, ascending sequence of faces, and set of barycenters of faces, convex combination of subset of set of barycenters is convex combination w.r.t. set of vertexes of affine simplex |
| 570: For Simplicial Complex, Vertex of Simplex That Is on Another Simplex Is Vertex of Latter Simplex |
| description/proof of that for simplicial complex, vertex of simplex that is on another simplex is vertex of latter simplex |
| 571: For Simplicial Complex, Simplex Interior of Maximal Simplex Does Not Intersect Any Other Simplex |
| description/proof of that for simplicial complex, simplex interior of maximal simplex does not intersect any other simplex |
| 572: When Convex Set Spanned by Non-Affine-Independent Set of Base Points on Real Vectors Space Is Affine Simplex, Point Whose Original Coefficients Are All Positive Is on Simplex Interior of Simplex, but Point One of Whose Original Coefficients Is 0 Is Not Necessarily on Simplex Boundary of Simplex |
| description/proof of that when convex set spanned by non-affine-independent set of base points on real vectors space is affine simplex, point whose original coefficients are all positive is on simplex interior of simplex, but point one of whose original coefficients is 0 is not necessarily on simplex boundary of simplex |
| 573: Domain of Affine Simplex Map Is Closed and Compact on Euclidean Topological Superspace |
| description/proof of that domain of affine simplex map is closed and compact on Euclidean topological superspace |
| 574: Affine Simplex Map into Finite-Dimensional Vectors Space Is Continuous w.r.t. Canonical Topologies |
| description/proof of that affine simplex map into finite-dimensional vectors space is continuous w.r.t. canonical topologies |
| 575: Affine Simplex on Finite-Dimensional Real Vectors Space Is Closed and Compact on Canonical Topological Superspace |
| description/proof of that affine simplex on finite-dimensional real vectors space is closed and compact on canonical topological superspace |
| 576: Simplex Interior of Affine Simplex Is Open on Affine Simplex with Canonical Topology |
| description/proof of that simplex interior of affine simplex is open on affine simplex with canonical topology |
| 577: Element of Simplicial Complex on Finite-Dimensional Real Vectors Space Is Closed and Compact on Underlying Space of Complex |
| description/proof of that element of simplicial complex on finite-dimensional real vectors space is closed and compact on underlying space of complex |
| 578: For Finite Simplicial Complex on Finite-Dimensional Real Vectors Space, Simplex Interior of Maximal Simplex Is Open on Underlying Space of Complex |
| description/proof of that for finite simplicial complex on finite-dimensional real vectors space, simplex interior of maximal simplex is open on underlying space of complex |
| 579: Vertex of Affine Simplex |
| definition of vertex of affine simplex |
| 580: Vertex in Simplicial Complex |
| definition of vertex in simplicial complex |
| 581: Star of Vertex in Simplicial Complex |
| definition of star of vertex in simplicial complex |
| 582: Intersection of Union of Subsets and Subset Is Union of Intersections of Each of Subsets and Latter Subset |
| description/proof of that intersection of union of subsets and subset is union of intersections of each of subsets and latter subset |
| 583: Subset of Underlying Space of Finite Simplicial Complex on Finite-Dimensional Real Vectors Space Is Closed iff Its Intersection with Each Element of Complex Is Closed |
| description/proof of that subset of underlying space of finite simplicial complex on finite-dimensional real vectors space is closed iff its intersection with each element of complex is closed |
| 584: For Simplicial Complex on Finite-Dimensional Real Vectors Space, Open Subset of Underlying Space That Intersects Star Intersects Simplex Interior of Maximal Simplex Involved in Star |
| description/proof of that for simplicial complex on finite-dimensional real vectors space, open subset of underlying space that intersects star intersects simplex interior of maximal simplex involved in star |
| 585: Boundary of Subset of Topological Space Is Set of Points of Each of Which Each Neighborhood Intersects Both Subset and Complement of Subset |
| description/proof of that boundary of subset of topological space is set of points of each of which each neighborhood intersects both subset and complement of subset |
| 586: For Topological Space, Subspace, and Subset of Superspace, Subspace Minus Subset as Subspace of Subspace Is Subspace of Superspace Minus Subset |
| description/proof of that for topological space, subspace, and subset of superspace, subspace minus subset as subspace of subspace is subspace of superspace minus subset |
| 587: Limit of Normed Vectors Spaces Map at Point |
| definition of limit of normed vectors spaces map at point |
| 588: Normed Vectors Spaces Map Continuous at Point |
| definition of normed vectors spaces map continuous at point |
| 589: Quotient Group of Group by Normal Subgroup |
| definition of quotient group of group by normal subgroup |
| 590: Group as Direct Sum of Finite Number of Normal Subgroups |
| definition of group as direct sum of finite number of normal subgroups |
| 591: Direct Product of Structures |
| definition of direct product of structures |
| 592: Direct Sum of Modules |
| definition of direct sum of modules |
| 593: Finite Product of Subgroups Is Associative |
| description/proof of that finite product of subgroups is associative |
| 594: Conjugate Subgroup of Subgroup by Element |
| definition of conjugate subgroup of subgroup by element |
| 595: For Group and Its Subgroup, Subgroup Is Normal Subgroup if Its Conjugate Subgroup by Each Element of Group Is Contained in It |
| description/proof of that for group and its subgroup, subgroup is normal subgroup if its conjugate subgroup by each element of group is contained in it |
| 596: Subgroup of Group Multiplied by Normal Subgroup of Group Is Subgroup of Group |
| description/proof of that subgroup of group multiplied by normal subgroup of group is subgroup of group |
| 597: Finite Product of Normal Subgroups Is Commutative and Is Normal Subgroup |
| description/proof of that finite product of normal subgroups is commutative and is normal subgroup |
| 598: Normal Subgroup of Group Is Normal Subgroup of Subgroup of Group Multiplied by Normal Subgroup |
| description/proof of that normal subgroup of group is normal subgroup of subgroup of group multiplied by normal subgroup |
| 599: Group as Direct Sum of Finite Number of Normal Subgroups Is Group as Direct Sum of Any Reordered and Combined Normal Subgroups |
| description/proof of that group as direct sum of finite number of normal subgroups is group as direct sum of any reordered and combined normal subgroups |
| 600: For Group as Direct Sum of Finite Number of Normal Subgroups, Element Is Uniquely Decomposed and Decomposition Is Commutative |
| description/proof of that for group as direct sum of finite number of normal subgroups, element is uniquely decomposed and decomposition is commutative |
| 601: Bijective Group Homomorphism Is 'Groups - Homomorphisms' Isomorphism |
| description/proof of that bijective group homomorphism is 'groups - homomorphisms' isomorphism |
| 602: Range of Group Homomorphism Is Subgroup of Codomain |
| description/proof of that range of group homomorphism is subgroup of codomain |
| 603: Injective Group Homomorphism Is 'Groups - Homomorphisms' Isomorphism onto Range |
| description/proof of that injective group homomorphism is 'groups - homomorphisms' isomorphism onto range |
| 604: Group as Direct Sum of Finite Number of Normal Subgroups Is 'Groups - Homomorphisms' Isomorphic to Direct Product of Subgroups |
| description/proof of that group as direct sum of finite number of normal subgroups is 'groups - homomorphisms' isomorphic to direct product of subgroups |
| 605: Finite Direct Product of Groups Is 'Groups - Homomorphisms' Isomorphic to Direct Product of Corresponding Isomorphic Groups |
| description/proof of that finite direct product of groups is 'groups - homomorphisms' isomorphic to direct product of corresponding isomorphic groups |
| 606: For 2 Decompositions of Vector with Common Constituent, Coefficients of Common Constituent Are Same if Common Constituent Is Not on Vectors Space Spanned by Other Constituents |
| description/proof of that for 2 decompositions of vector with common constituent, coefficients of common constituent are same if common constituent is not on vectors space spanned by other constituents |
| 607: Congruence on Category |
| definition of congruence on category |
| 608: Quotient Category |
| definition of quotient category |
| 609: Top Category |
| definition of Top category |
| 610: Top^2 Category |
| definition of \(Top^2\) category |
| 611: Top^Asterisk Category |
| definition of \(Top^*\) category |
| 612: Simplicial Map |
| definition of simplicial map |
| 613: 'Finite Simplicial Complexes - Simplicial Maps' Category |
| definition of 'finite simplicial complexes - simplicial maps' category |
| 614: Affine Map from Affine or Convex Set Spanned by Possibly-Non-Affine-Independent Set of Base Points on Finite-Dimensional Real Vectors Space into Finite Dimensional Real Vectors Space Is Continuous W.r.t. Canonical Topologies |
| description/proof of that affine map from affine or convex set spanned by possibly-non-affine-independent set of base points on finite-dimensional real vectors space into finite dimensional real vectors space is continuous w.r.t. canonical topologies |
| 615: 'Finite Simplicial Complexes - Simplicial Maps' Category to Top Functor |
| definition of 'finite simplicial complexes - simplicial maps' category to Top functor |
| 616: Retraction |
| definition of retraction |
| 617: Retract of Topological Space |
| definition of retract of topological space |
| 618: Deformation Retraction |
| definition of deformation retraction |
| 619: Deformation Retract of Topological Space |
| definition of deformation retract of topological space |
| 620: For Simplicial Complex, Intersection of 2 Simplexes Is Simplex Determined by Intersection of Sets of Vertexes of Simplexes |
| description/proof of that for simplicial complex, intersection of 2 simplexes is simplex determined by intersection of sets of vertexes of simplexes |
| 621: Ascending Sequence of Barycenters of Faces of Affine Simplex |
| definition of ascending sequence of barycenters of faces of affine simplex |
| 622: For Simplicial Complex, Intersection of 2 Affine Simplexes Determined by Subsequences of Ascending Sequences of Barycenters of Faces of Elements of Complex Is Affine Simplex Determined by Intersection of Subsequences |
| description/proof of that for simplicial complex, intersection of 2 affine simplexes determined by subsequences of ascending sequences of barycenters of faces of elements of complex is affine simplex determined by intersection of subsequences |
| 623: Barycentric Subdivision of Simplicial Complex |
| definition of barycentric subdivision of simplicial complex |
| 624: For Map, if There Is Inverse Direction Map Which After Original Map Is Identity, Original Map Is Injective |
| description/proof of that for map, if there is inverse direction map which after original map is identity, original map is injective |
| 625: For Linearly Independent Sequence in Vectors Space, Derived Sequence in Which Each Element Is Linear Combination of Equal or Smaller Index Elements with Nonzero Equal Index Coefficient Is Linearly Independent |
| description/proof of that for linearly independent sequence in vectors space, derived sequence in which each element is linear combination of equal or smaller index elements with nonzero equal index coefficient is linearly independent |
| 626: Basis of Module |
| definition of basis of module |
| 627: Generator of Module |
| definition of generator of module |
| 628: Finitely-Generated Module |
| definition of finitely-generated module |
| 629: Principal Ideal of Ring |
| definition of principal ideal of ring |
| 630: Integral Domain |
| definition of integral domain |
| 631: Principal Integral Domain |
| definition of principal integral domain |
| 632: For Simplicial Complex, Point on Underlying Space Is on Simplex Interior of Unique Simplex |
| description/proof of that for simplicial complex, point on underlying space is on simplex interior of unique simplex |
| 633: For Simplicial Complex, Stars of Vertexes of Simplexes Is Open Cover of Underlying Space |
| description/proof of that for finite simplicial complex, stars of vertexes of simplexes is open cover of underlying space |
| 634: Functor Maps Isomorphism to Isomorphism |
| description/proof of that functor maps isomorphism to isomorphism |
| 635: Range Under Lie Algebra Homomorphism Is Lie Sub-Algebra of Codomain |
| description/proof of that range under Lie algebra homomorphism is Lie sub-algebra of codomain |
| 636: Equivalence Relation on Set |
| definition of equivalence relation on set |
| 637: Quotient Set |
| definition of quotient set |
| 638: Representatives Set of Quotient Set |
| definition of representatives set of quotient set |
| 639: Units of Ring |
| definition of units of ring |
| 640: Associates of Element of Commutative Ring |
| definition of associates of element of commutative ring |
| 641: Greatest Common Divisors of Subset of Commutative Ring |
| definition of greatest common divisors of subset of commutative ring |
| 642: Least Common Multiples of Subset of Commutative Ring |
| definition of least common multiples of subset of commutative ring |
| 643: For Ring, Multiple of 0 Is 0 |
| description/proof of that for ring, multiple of 0 is 0 |
| 644: Cancellation Rule on Integral Domain |
| description/proof of cancellation rule on integral domain |
| 645: For Integral Domain, if Greatest Common Divisors of Subset Exist, They Are Associates of a Greatest Common Divisor |
| description/proof of that for integral domain, if greatest common divisors of subset exist, they are associates of a greatest common divisor |
| 646: For Integral Domain, if Least Common Multiples of Subset Exist, They Are Associates of a Least Common Multiple |
| description/proof of that for integral domain, if least common multiples of subset exist, they are associates of a least common multiple |
| 647: Irreducible Element of Commutative Ring |
| definition of irreducible element of commutative ring |
| 648: Unique Factorization Domain |
| definition of unique factorization domain |
| 649: For Unique Factorization Domain, if Multiple of Elements Is Divisible by Irreducible Element, at Least 1 Constituent Is Divisible by Irreducible Element |
| description/proof of that for unique factorization domain, if multiple of elements is divisible by irreducible element, at least 1 constituent is divisible by irreducible element |
| 650: For Unique Factorization Domain, Method of Getting Greatest Common Divisors of Finite Subset by Factorizing Each Element of Subset with Representatives Set of Associates Quotient Set |
| description/proof of for unique factorization domain, method of getting greatest common divisors of finite subset by factorizing each element of subset with representatives set of associates quotient set |
| 651: For Unique Factorization Domain, Method of Getting Least Common Multiples of Finite Subset by Factorizing Each Element of Subset with Representatives Set of Associates Quotient Set |
| description/proof of for unique factorization domain, method of getting least common multiples of finite subset by factorizing each element of subset with representatives set of associates quotient set |
| 652: For Unique Factorization Domain and Finite Subset, iff Greatest Common Divisors of Each Pair Subset of Subset Are Unit Associates, Least Common Multiples of Subset Are Associates of Multiple of Elements of Subset |
| description/proof of that for unique factorization domain and finite subset, iff greatest common divisors of each pair subset of subset are unit associates, least common multiples of subset are associates of multiple of elements of subset |
| 653: For Unique Factorization Domain and Finite Subset, if Greatest Common Divisors of Each Pair Subset of Subset Are Unit Associates, Greatest Common Divisors of Subset Are Unit Associates, but Not Vice Versa |
| description/proof of that for unique factorization domain and finite subset, if greatest common divisors of each pair subset of subset are unit associates, greatest common divisors of subset are unit associates, but not vice versa |
| 654: For Group as Direct Sum of Finite Number of Normal Subgroups, Product of Subset of Normal Subgroups Is Group as Direct Sum of Subset |
| description/proof of that for group as direct sum of finite number of normal subgroups, product of subset of normal subgroups is group as direct sum of subset |
| 655: Greatest Common Divisors Domain |
| definition of greatest common divisors domain |
| 656: For Ring and Finite Number of Ideals, Sum of Ideals Is Ideal |
| description/proof of that for ring and finite number of ideals, sum of ideals is ideal |
| 657: Principal Integral Domain Is Greatest Common Divisors Domain, and for 2 Elements, Each of Greatest Common Divisors Is One by Which Sum of Principal Ideals by 2 Elements Is Principal Ideal |
| description/proof of that principal integral domain is greatest common divisors domain, and for 2 elements, each of greatest common divisors is one by which sum of principal ideals by 2 elements is principal ideal |
| 658: For Commutative Ring, if Each Elements Pair Has Greatest Common Divisor, Each Finite Subset Has Greatest Common Divisor, Which Can Be Gotten Sequentially |
| description/proof of that for commutative ring, if each elements pair has greatest common divisor, each finite subset has greatest common divisor, which can be gotten sequentially |
| 659: For Integral Domain, if Principal Ideal by Element Is Also by Another Element, Elements Are Associates with Each Other, and Principal Ideal Is by Any Associate |
| description/proof of that for integral domain, if principal ideal by element is also by another element, elements are associates with each other, and principal ideal is by any associate |
| 660: For Principal Integral Domain and Finite Subset, Sum of Principal Ideals by Elements of Subset Is Principal Ideal by Any of Greatest Common Divisors of Subset |
| description/proof of that for principal integral domain and finite subset, sum of principal ideals by elements of subset is principal ideal by any of greatest common divisors of subset |
| 661: 6-Elements Group Cannot Have 2 3-Elements Subgroups That Share Only Identity |
| description/proof of that 6-elements group cannot have 2 3-elements subgroups that share only identity |
| 662: \(C^\infty\) Immersion |
| definition of \(C^\infty\) immersion |
| 663: C^\infty Submersion |
| definition of \(C^\infty\) submersion |
| 664: Section of Continuous Map |
| definition of section of continuous surjection |
| 665: Tangent Vectors Bundle over \(C^\infty\) Manifold with Boundary |
| definition of tangent vectors bundle over \(C^\infty\) manifold with boundary |
| 666: \(C^\infty\) Vectors Field on \(C^\infty\) Manifold with Boundary |
| definition of \(C^\infty\) vectors field on \(C^\infty\) manifold with boundary |
| 667: For Rectangle Matrix over Principal Integral Domain, There Are Some Types of Rows or Columns Operations Each of Which Can Be Expressed as Multiplication by Invertible Matrix from Left or Right |
| description/proof of that for rectangle matrix over principal integral domain, there are some types of rows or columns operations each of which can be expressed as multiplication by invertible matrix from left or right |
| 668: For Principal Integral Domain, Rectangle Matrix over Domain, and Square Matrix Over Domain, Sum of Principal Ideals by Specified-Dimensional Subdeterminants of Product Is Contained in Sum of Principal Ideals by Same-Dimensional Subdeterminants of Rectangle Matrix |
| description/proof of that for principal integral domain, rectangle matrix over domain, and square matrix over domain, sum of principal ideals by specified-dimensional subdeterminants of product is contained in sum of principal ideals by same-dimensional subdeterminants of rectangle matrix |
| 669: For Principal Integral Domain, Rectangle Matrix over Domain, and Invertible Square Matrix over Domain, Sum of Principal Ideals by Specified-Dimensional Subdeterminants of Product Is Sum of Principal Ideals by Same Dimensional Subdeterminants of Rectangle Matrix |
| description/proof of that for principal integral domain, rectangle matrix over domain, and invertible square matrix over domain, sum of principal ideals by specified-dimensional subdeterminants of product is sum of principal ideals by same dimensional subdeterminants of rectangle matrix |
| 670: Smith Normal Form Theorem for Rectangle Matrix over Principal Integral Domain |
| description/proof of that Smith normal form theorem for rectangle matrix over principal integral domain |
| 671: Power Set of Set |
| definition of power set of set |
| 672: Locally Compact Topological Space |
| definition of locally compact topological space |
| 673: Locally Finite Set of Subsets of Topological Space |
| definition of locally finite set of subsets of topological space |
| 674: Dimension of Vectors Space |
| definition of dimension of vectors space |
| 675: Complementary Subspace of Vectors Subspace |
| definition of complementary subspace of vectors subspace |
| 676: For Finite-Dimensional Vectors Space Basis, Replacing Element by Linear Combination of Elements with Nonzero Coefficient for Element Forms Basis |
| description/proof of that for finite dimensional vectors space basis, replacing element by linear combination of elements with nonzero coefficient for element forms basis |
| 677: For Finite-Dimensional Vectors Space, Linearly Independent Subset Can Be Expanded to Be Basis by Adding Finite Elements |
| description/proof of that for finite-dimensional vectors space, linearly independent subset can be expanded to be basis by adding finite elements |
| 678: For Finite-Dimensional Vectors Space, There Is No Basis That Has More Than Dimension Elements |
| description/proof of that for finite-dimensional vectors space, there is no basis that has more than dimension elements |
| 679: For Finite-Dimensional Vectors Space, There Is No Linearly Independent Subset That Has More Than Dimension Elements |
| description/proof of that for finite-dimensional vectors space, there is no linearly independent subset that has more than dimension elements |
| 680: For Finite-Dimensional Vectors Space, Proper Subspace Has Lower Dimension |
| description/proof of that for finite-dimensional vectors space, proper subspace has lower dimension |
| 681: For Finite-Dimensional Vectors Space, Linearly Independent Subset with Dimension Number of Elements Is Basis |
| description/proof of that for finite-dimensional vectors space, linearly independent subset with dimension number of elements is basis |
| 682: For Finite-Dimensional Vectors Space and Basis, Linearly Independent Set of Elements Can Be Augmented with Some Elements of Basis to Be Basis |
| description/proof of that for finite-dimensional vectors space and basis, linearly independent set of elements can be augmented with some elements of basis to be basis |
| 683: Polynomials Ring over Commutative Ring |
| definition of polynomials ring over commutative ring |
| 684: Polynomials Ring over Integral Domain Is Integral Domain |
| description/proof of that polynomials ring over integral domain is integral domain |
| 685: Over Field, Polynomial and Nonzero Polynomial Divisor Have Unique Quotient and Remainder |
| description/proof of that over field, polynomial and nonzero polynomial divisor have unique quotient and remainder |
| 686: Field Is Integral Domain |
| description/proof of that field is integral domain |
| 687: Over Field, n-Degree Polynomial Has at Most n Roots |
| description/proof of that over field, n-degree polynomial has at most n roots |
| 688: Integers Ring |
| definition of integers ring |
| 689: Integers Ring Is Principal Integral Domain |
| description/proof of that integers ring is principal integral domain |
| 690: Finite Composition of Injections Is Injection |
| description/proof of that finite composition of injections is injection |
| 691: Finite Composition of Surjections Is Not Necessarily Surjection |
| description/proof of that finite composition of surjections is not necessarily surjection |
| 692: Finite Composition of Surjections Is Surjection, if Codomains of Constituent Surjections Equal Domains of Succeeding Surjections |
| description/proof of that finite composition of surjections is surjection, if codomains of constituent surjections equal domains of succeeding surjections |
| 693: For Open Subset of d_1-Dimensional Euclidean C^\infty Manifold, C^\infty Map into d_2-Dimensional Euclidean C^\infty Manifold Divided by Never-Zero C^\infty Map into 1-Dimensional Euclidean C^\infty Manifold Is C^\infty |
| description/proof of that for open subset of \(d_1\)-dimensional Euclidean \(C^\infty\) manifold, \(C^\infty\) map into \(d_2\)-dimensional Euclidean \(C^\infty\) manifold divided by never-zero \(C^\infty\) map into 1-dimensional Euclidean \(C^\infty\) manifold is \(C^\infty\) |
| 694: Map Between Groups That Maps Product of 2 Elements to Product of Images of Elements Is Group Homomorphism |
| description/proof of that map between groups that maps product of 2 elements to product of images of elements is group homomorphism |
| 695: Memorandum on Powers of Group, Ring, or Field Elements |
| description/proof of memorandum on powers of group, ring, or field elements |
| 696: Conjugation for Group by Element |
| definition of conjugation for group by element |
| 697: For Group, Conjugation by Element Is 'Groups - Homomorphisms' Isomorphism |
| description/proof of that for group, conjugation by element is 'groups - homomorphisms' isomorphism |
| 698: Exhaustion of Topological Space by Compact Subsets |
| definition of exhaustion of topological space by compact subsets |
| 699: Exhaustion Function on Topological Space |
| definition of exhaustion function on topological space |
| 700: For Topological Space, Sequence of Preimages of Natural-Numbers-Closed-Upper-Bounds Intervals Under Exhaustion Function Is Exhaustion of Space by Compact Subsets |
| description/proof of that for topological space, sequence of preimages of natural-numbers-closed-upper-bounds intervals under exhaustion function is exhaustion of space by compact subsets |
| 701: Chart on \(C^\infty\) Manifold with Boundary |
| definition of chart on \(C^\infty\) manifold with boundary |
| 702: Chart Ball Around Point on \(C^\infty\) Manifold with Boundary |
| definition of chart ball around point on \(C^\infty\) manifold with boundary |
| 703: Chart Half Ball Around Point on \(C^\infty\) Manifold with Boundary |
| definition of chart half ball around point on \(C^\infty\) manifold with boundary |
| 704: For \(C^\infty\) Manifold with Boundary, Interior Point Has Chart Ball and Boundary Point Has Chart Half Ball |
| description/proof of that for \(C^\infty\) manifold with boundary, interior point has chart ball and boundary point has chart half ball |
| 705: For Euclidean \(C^\infty\) Manifold, Open Ball Is Diffeomorphic to Whole Space |
| description/proof of that for Euclidean \(C^\infty\) manifold, open ball is diffeomorphic to whole space |
| 706: For Half Euclidean \(C^\infty\) Manifold with Boundary, Open Half Ball Is Diffeomorphic to Whole Space |
| description/proof of that for half Euclidean \(C^\infty\) manifold with boundary, open half ball is diffeomorphic to whole space |
| 707: For \(C^\infty\) Manifold with Boundary, Interior Point Has Chart Whose Range Is Whole Euclidean Space and Boundary Point Has Chart Whose Range Is Whole Half Euclidean Space |
| description/proof of that for \(C^\infty\) manifold with boundary, interior point has chart whose range is whole Euclidean space and boundary point has chart whose range is whole half Euclidean space |
| 708: For Map from Subset of \(C^\infty\) Manifold with Boundary into Subset of \(C^\infty\) Manifold \(C^k\) at Point, There Is \(C^k\) Extension on Open-Neighborhood-of-Point Domain |
| description/proof of that for map from subset of \(C^\infty\) manifold with boundary into subset of \(C^\infty\) manifold \(C^k\) at point, there is \(C^k\) extension on-open-neighborhood-of-point domain |
| 709: For Map from Subset of \(C^\infty\) Manifold with Boundary into Subset of \(C^\infty\) Manifold with Boundary, Map Is Local Diffeomorphism iff for Each Domain Point and Its Image, There Are Charts by Which Coordinates Function Is Diffeomorphism |
| description/proof of that for map from subset of \(C^\infty\) manifold with boundary into subset of \(C^\infty\) manifold with boundary, map is local diffeomorphism iff for each domain point and its image, there are charts by which coordinates function is diffeomorphism |
| 710: n-Symmetric Group |
| definition of n-symmetric group |
| 711: m-Cycle on n-Symmetric Group |
| definition of m-cycle on n-symmetric group |
| 712: Cyclic Group by Element |
| definition of cyclic group by element |
| 713: Centralizer of Element on Group |
| definition of centralizer of element on group |
| 714: For n-Symmetric Group and n-Cycle, Centralizer of Cycle on Symmetric Group Is Cyclic Group by Cycle |
| description/proof of that for n-symmetric group and n-cycle, centralizer of cycle on symmetric group is cyclic group by cycle |
| 715: Proposition 1 or Proposition 2 iff if Not Proposition 2, Proposition 1 |
| description/proof of that proposition 1 or proposition 2 iff if not proposition 2, proposition 1 |
| 716: For Group, Normal Subgroup, and Subgroup, Subsets of Quotient Group That Contain Cosets of Subgroup Are Same or Disjoint |
| description/proof of that for group, normal subgroup, and subgroup, subsets of quotient group that contain cosets of subgroup are same or disjoint |
| 717: For Group, Multiplication Map with Fixed Element from Left or Right Is Bijection |
| description/proof of that for group, multiplication map with fixed element from left or right is bijection |
| 718: For Group, Normal Subgroup, and Quotient Group, Representatives Set Multiplied by Element Is Representatives Set |
| description/proof of that for group, normal subgroup, and quotient group, representatives set multiplied by element is representatives set |
| 719: For Linear Map from Finite-Dimensional Vectors Space, There Is Domain Subspace That Is 'Vectors Spaces - Linear Morphisms' Isomorphic to Range by Restriction of Map |
| description/proof of that for linear map from finite-dimensional vectors space, there is domain subspace that is 'vectors Spaces - linear morphisms' isomorphic to range by restriction of map |
| 720: For Finite-Dimensional Vectors Space, Subset That Spans Space Can Be Reduced to Be Basis |
| description/proof of that for finite-dimensional vectors space, subset that spans space can be reduced to be basis |
| 721: For Vectors Space, Intersection of Finite-Dimensional Subspaces Is Subspace with Dimension Equal to or Smaller than Minimum Dimension of Subspaces |
| description/proof of that for vectors space, intersection of finite-dimensional subspaces is subspace with dimension equal to or smaller than minimum dimension of subspaces |
| 722: For Vectors Space and 2 Same-Finite-Dimensional Vectors Subspaces, There Is Common Complementary Subspace |
| description/proof of that for vectors space and 2 same-finite-dimensional vectors subspaces, there is common complementary subspace |
| 723: Reversed Operator Group of Group |
| definition of reversed operator group of group |
| 724: Group Is 'Groups - Homomorphisms' Isomorphic to Reversed Operator Group of Group |
| description/proof of that group is 'groups - homomorphisms' isomorphic to reversed operator group of group |
| 725: For Vectors Space, Generator of Space, and Linearly Independent Subset Contained in Generator, Generator Can Be Reduced to Be Basis with Linearly Independent Subset Retained |
| description/proof of that for vectors space, generator of space, and linearly independent subset contained in generator, generator can be reduced to be basis with linearly independent subset retained |
| 726: For Vectors Space and Linearly Independent Subset, Subset Can Be Expanded to Be Basis |
| description/proof of that for vectors space and linearly independent subset, subset can be expanded to be basis |
| 727: For Vectors Space, Finite Generator Can Be Reduced to Be Basis |
| description/proof of that for vectors space, finite generator can be reduced to be basis |
| 728: For Linear Surjection Between Finite-Dimensional Vectors Spaces, Dimension of Codomain Is Equal to or Smaller than That of Domain |
| description/proof of that for linear surjection between finite-dimensional vectors spaces, dimension of codomain is equal to or smaller than that of domain |
| 729: For Linear Surjection from Finite-Dimensional Vectors Space, if Dimension of Codomain Is Equal to or Larger than That of Domain, Surjection Is Bijection |
| description/proof of that for linear surjection from finite-dimensional vectors space, if dimension of codomain is equal to or larger than that of domain, surjection is bijection |
| 730: Union of Set |
| definition of union of set |
| 731: For Set, Union of Power Set of Set Is Set |
| description/proof of that for set, union of power set of set is set |
| 732: Latin Square of Finite Set |
| definition of Latin square of finite set |
| 733: Latin Square with Each Row Regarded as Permutation Forms Group iff Composition of 2 Rows Is Row, and Group's Multiplications Table Is Generated by Certain Way from Square |
| description/proof of that Latin square with each row regarded as permutation forms group iff composition of 2 rows is row, and group's multiplications table is generated by certain way from square |
| 734: Set Elements Minus Set |
| definition of set elements minus set |
| 735: For Set and Set, Power Set of [Former Set Minus Latter Set] Is [Power Set of Former Set] Elements Minus Latter Set |
| description/proof of that for set and set, power set of [former set minus latter set] is [power set of former set] elements minus latter set |
| 736: Projection of Vector into Vectors Subspace w.r.t. Complementary Subspace |
| definition of projection of vector into vectors subspace w.r.t. complementary subspace |
| 737: Projection from Vectors Space into Subspace w.r.t. Complementary Subspace Is Linear Map and Image of Any Subspace Is Subspace |
| description/proof of that projection from vectors space into subspace w.r.t. complementary subspace is linear map and image of any subspace is subspace |
| 738: For Vectors Space, Subspace, and Complementary Subspace, Finite-Dimensional Subspace That Intersects Complementary Subspace Trivially Is Projected into Subspace as Same-Dimensional Subspace |
| description/proof of that for vectors space, subspace, and complementary subspace, finite-dimensional subspace that intersects complementary subspace trivially is projected into subspace as same-dimensional subspace |
| 739: Motion |
| definition of motion |
| 740: Orthogonal Linear Map |
| definition of orthogonal linear map |
| 741: Orthogonal Linear Map Is Motion |
| description/proof of that orthogonal linear map is motion |
| 742: Motion Is Injective |
| description/proof of that motion is injective |
| 743: Linear Injection Between Same-Finite-Dimensional Vectors Spaces Is 'Vectors Spaces - Linear Morphisms' Isomorphism |
| description/proof of that linear injection between same-finite-dimensional vectors spaces is 'vectors spaces - linear morphisms' isomorphism |
| 744: Orthogonal Linear Map Between Same-Finite-Dimensional Normed Vectors Spaces Is 'Vectors Spaces - Linear Morphisms' Isomorphism and Inverse Is Orthogonal Linear Map |
| description/proof of that orthogonal linear map between same-finite-dimensional normed vectors spaces is 'vectors spaces - linear morphisms' isomorphism and inverse is orthogonal linear map |
| 745: For Vectors Space with Inner Product, Set of Nonzero Orthogonal Elements Is Linearly Independent |
| description/proof of that for vectors space with inner product, set of nonzero orthogonal elements is linearly independent |
| 746: For Motion Between Real Vectors Spaces with Norms Induced by Inner Products That Fixes 0, Orthonormal Subset of Domain Is Mapped to Orthonormal Subset |
| description/proof of that for motion between real vectors spaces with norms induced by inner products that fixes 0, orthonormal subset of domain is mapped to orthonormal subset |
| 747: For Motion Between Same-Finite-Dimensional Real Vectors Spaces with Norms Induced by Inner Products That Fixes 0, Motion Is Orthogonal Linear Map |
| description/proof of that for motion between same-finite-dimensional real vectors spaces with norms induced by inner products that fixes 0, motion is orthogonal linear map |
| 748: Finite Composition of Motions Is Motion |
| description/proof of that finite composition of motions is motion |
| 749: Finite Composition of Bijections Is Bijection, if Codomains of Constituent Bijections Equal Domains of Succeeding Bijections |
| description/proof of that finite composition of bijections is bijection, if codomains of constituent bijections equal domains of succeeding bijections |
| 750: For Motion Between Same-Finite-Dimensional Real Vectors Spaces with Norms Induced by Inner Products, Motion Is Bijective |
| description/proof of that for motion between same-finite-dimensional real vectors spaces with norms induced by inner products, motion is bijective |
| 751: For Finite-Dimensional Normed Real Vectors Space with Canonical Topology, Norm Map Is Continuous |
| description/proof of that for finite-dimensional normed real vectors space with canonical topology, norm map is continuous |
| 752: Group Action |
| definition of group action |
| 753: For Group Action, Induced Map with Fixed Group Element Is Bijection |
| description/proof of that for group action, induced map with fixed group element is bijection |
| 754: For Euclidean Topological Space, Lower-Dimensional Euclidean Topological Space, Slicing Map, Projection, and Inclusion, Inclusion after Projection after Slicing Map Equals Slicing Map, and Projection after Slicing Map of Open Neighborhood of Point Is Open Neighborhood of Projection of Point |
| description/proof of that for Euclidean topological space, lower-dimensional Euclidean topological space, slicing map, projection, and inclusion, inclusion after projection after slicing map equals slicing map, and projection after slicing map of open neighborhood of point is open neighborhood of projection of point |
| 755: For Nonzero Linear Map Between Normed Vectors Spaces, Image Norm Divided by Argument Norm Does Not Converge to 0 When Argument Norm Nears 0 |
| description/proof of that for nonzero linear map between normed vectors spaces, image norm divided by argument norm does not converge to 0 when argument norm nears 0 |
| 756: For Map Between Normed Vectors Spaces s.t. Image Norm Divided by Argument Norm Converges to 0 When Argument Norm Nears 0, Image Norm of Map Plus Nonzero Linear Map Divided by Argument Norm Does Not Do So |
| description/proof of that for map between normed vectors spaces s.t. image norm divided by argument norm converges to 0 when argument norm nears 0, image norm of map plus nonzero linear map divided by argument norm does not do so |
| 757: Equivalence Relation on Norms on Vectors Space |
| definition of equivalence relation on norms on vectors space |
| 758: Norms on Finite-Dimensional Real Vectors Space Are Equivalent |
| description/proof of that norms on finite-dimensional real vectors space are equivalent |
| 759: Product of Hausdorff Topological Spaces Is Hausdorff |
| description/proof of that product of Hausdorff topological spaces is Hausdorff |
| 760: Finite Product of 2nd-Countable Topological Spaces Is 2nd-Countable |
| description/proof of that finite product of 2nd-countable topological spaces is 2nd-countable |
| 761: Euclidean Topological Space Is Homeomorphic to Product of Lower-Dimensional Euclidean Spaces |
| description/proof of that Euclidean topological space is homeomorphic to product of lower-dimensional Euclidean spaces |
| 762: Closed Upper Half Euclidean Topological Space Is Homeomorphic to Product of Lower-Dimensional Euclidean Spaces and Closed Upper Half Euclidean Space |
| description/proof of that closed upper half Euclidean topological space is homeomorphic to product of lower-dimensional Euclidean spaces and closed upper half Euclidean space |
| 763: Composition of Product Maps Is Product of Compositions of Component Maps |
| description/proof of that composition of product maps is product of compositions of component maps |
| 764: Finite-Product \(C^\infty\) Manifold with Boundary |
| definition of finite-product \(C^\infty\) manifold with boundary |
| 765: For \(C^\infty\) Map from Finite-Product \(C^\infty\) Manifold with Boundary, Induced Map with Set of Components of Domain Fixed Is \(C^\infty\) |
| description/proof of that for \(C^\infty\) map from finite-product \(C^\infty\) manifold with boundary, induced map with set of components of domain fixed is \(C^\infty\) |
| 766: For Continuous Map from Product Topological Space into Topological Space, Induced Map with Set of Components of Domain Fixed Is Continuous |
| description/proof of that for continuous map from product topological space into topological space, induced map with set of components of domain fixed is continuous |
| 767: Slicing Map on Euclidean Set |
| definition of slicing map on Euclidean set |
| 768: Slicing-And-Halving Map on Euclidean Set |
| definition of slicing-and-halving map on Euclidean set |
| 769: Curve on Topological Space |
| definition of curve on topological space |
| 770: Velocity of \(C^\infty\) Curve at Point on \(C^\infty\) Manifold with Boundary |
| definition of velocity of \(C^\infty\) curve at point on \(C^\infty\) manifold with boundary |
| 771: Tangent Vector at Point on \(C^\infty\) Manifold with Boundary Is Velocity of \(C^\infty\) Curve, Especially from Half Closed Interval, Especially as Linear in Coordinates |
| description/proof of that tangent vector at point on \(C^\infty\) manifold with boundary is velocity of \(C^\infty\) curve, especially from half closed interval, especially as linear in coordinates |
| 772: Map Is Bijection iff Preimage of Codomain Point Is 1 Point Subset |
| description/proof of that map is bijection iff preimage of codomain point is 1 point subset |
| 773: Open Ball Around Point on Metric Space |
| definition of open ball around point on metric space |
| 774: Formalization of Local Slice Condition for Embedded Submanifold or Local Slice Condition for Embedded Submanifold with Boundary |
| description/proof of formalization of local slice condition for embedded submanifold or local slice condition for embedded submanifold with boundary |
| 775: Bijective Linear Map Is 'Vectors Spaces - Linear Morphisms' Isomorphism |
| description/proof of that bijective linear map is 'vectors spaces - linear morphisms' isomorphism |
| 776: Bijective Lie Algebra Homomorphism Is 'Lie Algebras - Homomorphisms' Isomorphism |
| description/proof of that bijective Lie algebra homomorphism is 'Lie algebras - homomorphisms' isomorphism |
| 777: Embedded Submanifold with Boundary of \(C^\infty\) Manifold with Boundary |
| definition of embedded submanifold with boundary of \(C^\infty\) manifold with boundary |
| 778: Proper Map |
| definition of proper map |
| 779: Properly Embedded Submanifold with Boundary of \(C^\infty\) Manifold with Boundary |
| definition of properly embedded submanifold with boundary of \(C^\infty\) manifold with boundary |
| 780: Regular Domain of \(C^\infty\) Manifold with Boundary |
| definition of regular domain of \(C^\infty\) manifold with boundary |
| 781: For \(C^\infty\) Manifold with Boundary and Regular Domain, Differential of Inclusion at Point on Regular Domain Is 'Vectors Spaces - Linear Morphisms' Isomorphism |
| description/proof of that for \(C^\infty\) manifold with boundary and regular domain, differential of inclusion at point on regular domain is 'vectors spaces - linear morphisms' isomorphism |
| 782: For \(C^\infty\) Manifold, Embedded Submanifold with Boundary, and \(C^\infty\) Vectors Field over Submanifold with Boundary, Differential by Inclusion After Vectors Field Is \(C^\infty\) over Submanifold with Boundary |
| description/proof of that for \(C^\infty\) manifold, embedded submanifold with boundary, and \(C^\infty\) vectors field over submanifold with boundary, differential by inclusion after vectors field is \(C^\infty\) over submanifold with boundary |
| 783: For \(C^\infty\) Manifold, Subset, and Point on Subset, if Chart Satisfies Local Slice Condition for Embedded Submanifold or Local Slice Condition for Embedded Submanifold with Boundary, Its Sub-Open-Neighborhood Does So |
| description/proof of that for \(C^\infty\) manifold, subset, and point on subset, if chart satisfies local slice condition for embedded submanifold or local slice condition for embedded submanifold with boundary, its sub-open-neighborhood does so |
| 784: For \(C^\infty\) Manifold, Regular Domain, \(C^\infty\) Manifold with Boundary, and \(C^\infty\) Map from Regular Domain into \(C^\infty\) Manifold with Boundary, Corresponding Map with Domain Regarded as Subset of Manifold Is \(C^\infty\) |
| description/proof of that for \(C^\infty\) manifold, regular domain, \(C^\infty\) manifold with boundary, and \(C^\infty\) map from regular domain into \(C^\infty\) manifold with boundary, corresponding map with domain regarded as subset of manifold is \(C^\infty\) |
| 785: Simple Group |
| definition of simple group |
| 786: Abelian Group Is Simple Group iff Its Order Is Prime Number |
| description/proof of that Abelian group is simple group iff its order is prime number |
| 787: Subgroup Generated by Subset of Group |
| definition of subgroup generated by subset of group |
| 788: Order of Element of Group |
| definition of order of element of group |
| 789: For Group, Powers Sequence of Element That Returns Back Returns to Element |
| description/proof of that for group, powers sequence of element that returns back returns to element |
| 790: For Group and Finite-Order Element, Order Power of Element Is \(1\) and Subgroup Generated by Element Consists of Element to Non-Negative Powers Smaller Than Element Order |
| description/proof of that for group and finite-order element, order power of element is \(1\) and subgroup generated by element consists of element to non-negative powers smaller than element order |
| 791: For Group and Element, if There Is Positive Natural Number to Power of Which Element Is 1 and There Is No Smaller Such, Subgroup Generated by Element Consists of Element to Non-Negative Powers Smaller Than Number |
| description/proof of that for group and element, if there is positive natural number to power of which element is 1 and there is no smaller such, subgroup generated by element consists of element to non-negative powers smaller than number |
| 792: For Group and Finite-Order Element, Conjugate of Element Has Order of Element |
| description/proof of that for group and finite-order element, conjugate of element has order of element |
| 793: For Group and Finite-Order Element, Inverse of Element Has Order of Element |
| description/proof of that for group and finite-order element, inverse of element has order of element |
| 794: For Group and Element, if There Is Positive Natural Number to Power of Which Element Is 1 and There Is No Smaller Such, Integers of Which Powers to Which Element Are 1 Are Only Multiples of Number |
| description/proof of that for group and element, if there is positive natural number to power of which element is 1 and there is no smaller such, integers of which powers to which element are 1 are only multiples of number |
| 795: Intersection of Subgroup of Group and Normal Subgroup of Group Is Normal Subgroup of Subgroup |
| description/proof of that intersection of subgroup of group and normal subgroup of group is normal subgroup of subgroup |
| 796: For Topological Space Contained in Ambient Topological Space, if Space Is Ambient-Space-Wise Locally Topological Subspace of Ambient Space, Space Is Topological Subspace of Ambient Space |
| description/proof of that for topological space contained in ambient topological space, if space is ambient-space-wise locally topological subspace of ambient space, space is topological subspace of ambient space |
| 797: For Topological Space and Open Cover, Subset Is Open iff Intersection of Subset and Each Element of Open Cover Is Open |
| description/proof of that for topological space and open cover, subset is open iff intersection of subset and each element of open cover is open |
| 798: For Set and 2 Topologies, iff There Is Common Open Cover and Each Open Subset of Each Element of Cover in One Topology Is Open in the Other and Vice Versa, Topologies Are Same |
| description/proof of that for set and 2 topologies, iff there is common open cover and each open subset of each element of cover in one topology is open in the other and vice versa, topologies are same |
| 799: \(C^\infty\) Locally Trivial Surjection of Rank \(k\) |
| definition of \(C^\infty\) locally trivial surjection of rank \(k\) |
| 800: Closed Upper Half Euclidean Topological Space |
| definition of closed upper half Euclidean Topological Space |
| 801: Closed Upper Half Euclidean \(C^\infty\) Manifold with Boundary |
| definition of closed upper half Euclidean \(C^\infty\) manifold with boundary |
| 802: Immersed Submanifold with Boundary of \(C^\infty\) Manifold with Boundary |
| definition of immersed submanifold with boundary of \(C^\infty\) manifold with boundary |
| 803: For Topological Space and Locally Finite Set of Closed Subsets, Union of Set Is Closed |
| description/proof of that for topological space and locally finite set of closed subsets, union of set is closed |
| 804: For Topological Space and Finite Number of Open Covers, Intersection of Covers Is Open Cover |
| description/proof of that for topological space and finite number of open covers, intersection of covers is open cover |
| 805: For \(C^\infty\) Manifold with Boundary and Chart, Restriction of Chart on Open Subset Domain Is Chart |
| description/proof of that for \(C^\infty\) manifold with boundary and chart, restriction of chart on open subset domain is chart |
| 806: For Finite-Dimensional Vectors Space and Basis, Vectors Space Is 'Vectors Spaces - Linear Morphisms' Isomorphic to Components Vectors Space |
| description/proof of that for finite-dimensional vectors space and basis, vectors space is 'vectors spaces - linear morphisms' isomorphic to components vectors space |
| 807: For 'Vectors Spaces - Linear Morphisms' Isomorphism, Image of Linearly Independent Subset or Basis of Domain Is Linearly Independent or Basis on Codomain |
| description/proof of that for 'vectors spaces - linear morphisms' isomorphism, image of linearly independent subset or basis of domain is linearly independent or basis on codomain |
| 808: n-Cube Centered at \(p\) with Edges-Length \(l\) with Indexes \(B\) |
| definition of \(n\)-cube centered at \(p\) with edges-length \(l\) with indexes \(B\) |
| 809: n-Disk Centered at \(p\) with Radius \(r\) with Indexes \(B\) |
| definition of \(n\)-disk centered at \(p\) with radius \(r\) with indexes \(B\) |
| 810: Interior \(C^\infty\) Manifold of \(C^\infty\) Manifold with Boundary |
| definition of interior \(C^\infty\) manifold of \(C^\infty\) manifold with boundary |
| 811: Permutation Bijectively Maps Set of Permutations onto Set of Permutations by Composition from Left or Right |
| description/proof of that permutation bijectively maps set of permutations onto set of permutations by composition from left or right |
| 812: Same-Length Multi-Dimensional Array Antisymmetrized with Respect to Set of Indexes |
| definition of same-length multi-dimensional array antisymmetrized with respect to set of indexes |
| 813: Same-Length Multi-Dimensional Array Symmetrized with Respect to Set of Indexes |
| definition of same-length multi-dimensional array symmetrized with respect to set of indexes |
| 814: For Sequence of Finite Elements, Set of Permutations Has Same Number of Even Permutations and Odd Permutations |
| description/proof of for sequence of finite elements, set of permutations has same number of even permutations and odd permutations |
| 815: Antisymmetrized After Symmetrized Same-Length Multi-Dimensional Array or Symmetrized After Antisymmetrized Same-Length Multi-Dimensional Array Is 0 |
| description/proof of that antisymmetrized after symmetrized same-length multi-dimensional array or symmetrized after antisymmetrized same-length multi-dimensional array is 0 |
| 816: Contraction of 2 Same-Length Multi-Dimensional Arrays One of Which Is Symmetrized or Antisymmetrized w.r.t. Set of Indexes Is Contraction with Also Other Array Symmetrized or Antisymmetrized Accordingly |
| description/proof of that contraction of 2 same-length multi-dimensional arraysame-length multi-dimensional arrays one of which is symmetrized or antisymmetrized w.r.t. set of indexes is contraction with also other array symmetrized or antisymmetrized accordingly |
| 817: Neighborhood of Subset |
| definition of neighborhood of subset |
| 818: Support of Map from Topological Space into Field |
| definition of support of map from topological space into field |
| 819: Vectors Bundle of Rank \(k\) |
| definition of vectors bundle of rank \(k\) |
| 820: Intersection of Set |
| definition of intersection of set |
| 821: For \(C^\infty\) Embedding Between \(C^\infty\) Manifolds with Boundary, Restriction of Embedding on Embedded Submanifold with Boundary Domain Is \(C^\infty\) Embedding |
| description/proof of that for \(C^\infty\) embedding between \(C^\infty\) manifolds with boundary, restriction of embedding on embedded submanifold with boundary domain is \(C^\infty\) embedding |
| 822: For \(C^\infty\) Manifold with Boundary, Embedded Submanifold with Boundary of Embedded Submanifold with Boundary Is Embedded Submanifold with Boundary of Manifold with Boundary |
| description/proof of that for \(C^\infty\) manifold with boundary, embedded submanifold with boundary of embedded submanifold with boundary is embedded submanifold with boundary of manifold with boundary |
| 823: For \(C^\infty\) Manifold with Boundary and Embedded Submanifold with Boundary, Inverse of Codomain Restricted Inclusion Is \(C^\infty\) |
| description/proof of that for \(C^\infty\) manifold with boundary and embedded submanifold with boundary, inverse of codomain restricted inclusion is \(C^\infty\) |
| 824: For Map Between Embedded Submanifolds with Boundary of \(C^\infty\) Manifolds with Boundary, \(C^\infty\)-ness Does Not Change When Domain or Codomain Is Regarded to Be Subset |
| description/proof of that for map between embedded submanifolds with boundary of \(C^\infty\) manifolds with boundary, \(C^k\)-ness does not change when domain or codomain is regarded to be subset |
| 825: Section Along Subset of Codomain of Continuous Surjection |
| definition of section along subset of codomain of continuous surjection |
| 826: Trivializing Open Subset and Local Trivialization |
| definition of trivializing open subset and local trivialization |
| 827: \(C^\infty\) Trivializing Open Subset and \(C^\infty\) Local Trivialization |
| definition of \(C^\infty\) trivializing open subset and \(C^\infty\) local trivialization |
| 828: Open Subset of \(C^\infty\) Trivializing Open Subset Is \(C^\infty\) Trivializing Open Subset |
| description/proof of that open subset of \(C^\infty\) trivializing open subset is \(C^\infty\) trivializing open subset |
| 829: Open Submanifold with Boundary of \(C^\infty\) Manifold with Boundary |
| definition of open submanifold with boundary of \(C^\infty\) manifold with boundary |
| 830: For \(C^\infty\) Manifold with Boundary and Open Submanifold with Boundary, Differential of Inclusion at Point on Open Submanifold with Boundary Is 'Vectors Spaces- Linear Morphisms' Isomorphism |
| description/proof of that for \(C^\infty\) manifold with boundary and open submanifold with boundary, differential of inclusion at point on open submanifold with boundary is 'vectors spaces- linear morphisms' isomorphism |
| 831: Subbasis of Topological Space |
| definition of subbasis of topological space |
| 832: In Order to Check Continuousness of Map, Preimages of Only Basis or Subbasis Are Enough |
| description/proof of that in order to check continuousness of map, preimages of only basis or subbasis are enough |
| 833: For Vectors Bundle and Trivializing Open Subsets Cover, Preimages Under Trivializations of Products of Basis of Open Subset and Basis of \(R^k\) Constitute Basis of Total Space |
| description/proof of that for vectors bundle and trivializing open subsets cover, preimages under trivializations of products of basis of open subset and basis of \(R^k\) constitute basis of total space |
| 834: Matrices Multiplications Map Is Continuous |
| description/proof of that matrices multiplications map is continuous |
| 835: For n x n Matrix, if There Are m Rows with More Than n - m Same Columns 0, Matrix Is Not Invertible |
| description/proof of that for n x n matrix, if there are m rows with more than n - m same columns 0, matrix is not invertible |
| 836: For Invertible Square Matrix, from Top Row Downward Through Any Row, Each Row Can Be Changed to Have 1 1 Component and 0 Others Without Duplication to Keep Matrix Invertible |
| description/proof of that for invertible square matrix, from top row downward through any row, each row can be changed to have 1 1 component and 0 others without duplication to keep matrix invertible |
| 837: Map from Topological Space into Finite Product Topological Space Is Continuous iff All Component Maps Are Continuous |
| description/proof of that map from topological space into finite product topological space is continuous iff all component maps are continuous |
| 838: For Topological Space and Its 2 Products with Euclidean Topological Spaces, Map Between Products Fiber-Preserving and Linear on Fiber Is Continuous iff Canonical Matrix Is Continuous |
| description/proof of that for topological space and its 2 products with Euclidean topological spaces, map between products fiber-preserving and linear on fiber is continuous iff canonical matrix is continuous |
| 839: For Topological Space and Its 2 Products with Euclidean Topological Spaces, Injective Continuous Map Between Products Fiber-Preserving and Linear on Fiber Is Continuous Embedding |
| description/proof of that for topological space and its 2 products with Euclidean topological spaces, injective continuous map between products fiber-preserving and linear on fiber is continuous embedding |
| 840: Local \(C^\infty\) Frame on \(C^\infty\) Vectors Bundle |
| definition of local \(C^\infty\) frame on \(C^\infty\) vectors bundle |
| 841: p-Group |
| definition of p-group |
| 842: Center of Group |
| definition of center of group |
| 843: For Group and Subgroup, Conjugation for Subgroup by Group Element Is 'Groups Homomorphisms' Isomorphism |
| description/proof of that for group and subgroup, conjugation for subgroup by group element is 'groups homomorphisms' isomorphism |
| 844: For Group, Subgroup, and Element of Group, if \(k\) Is 1st Positive Power to Which Element Belongs to Subgroup, Multiples of \(k\) Are Only Powers to Which Element Belongs to Subgroup |
| description/proof of that for group, subgroup, and element of group, if \(k\) is 1st positive power to which element belongs to subgroup, multiples of \(k\) are only powers to which element belongs to subgroup |
| 845: For Finite \(p\)-Group, for Natural Number Smaller Than Power to Which \(p\) Is Order of Group, There Is Normal Subgroup of Group Whose Order Is \(p\) to Power of Natural Number |
| description/proof of that for finite \(p\)-group, for natural number smaller than power to which \(p\) is order of group, there is normal subgroup of group whose order is \(p\) to power of natural number |
| 846: For Group and Normal Subgroup, if Normal Subgroup and Quotient of Group by Normal Subgroup Are p-Groups, Group Is p-Group |
| description/proof of that for group and normal subgroup, if normal subgroup and quotient of group by normal subgroup are p-groups, group is p-group |
| 847: For (n + n') x (n + n'') Injective Matrix with Right-Top n x n'' Submatrix 0, Matrix with Left-Top n x n Submatrix Replaced with Injective Matrix Is Injective |
| description/proof of that for (n + n') x (n + n'') injective matrix with right-top n x n'' submatrix 0, matrix with left-top n x n submatrix replaced with injective matrix is injective |
| 848: For Maps Between Arbitrary Subspaces of Topological Spaces Continuous at Corresponding Points, Composition Is Continuous at Point |
| description/proof of that for maps between arbitrary subspaces of topological spaces continuous at corresponding points, composition is continuous at point |
| 849: Composition of \(C^\infty\) Embedding After Diffeomorphism or Diffeomorphism After \(C^\infty\) Embedding Is \(C^\infty\) Embedding |
| description/proof of that composition of \(C^\infty\) embedding after diffeomorphism or diffeomorphism after \(C^\infty\) embedding is \(C^\infty\) embedding |
| 850: For Topological Space, Point, and Neighborhood of Point, Neighborhood of Point on Neighborhood Is Neighborhood of Point on Base Space |
| description/proof of that for topological space, point, and neighborhood of point, neighborhood of point on neighborhood is neighborhood of point on base space |
| 851: For Covering Map, Cardinalities of Sheets Are Same |
| description/proof of that for covering map, cardinalities of sheets are same |
| 852: For \(C^\infty\) Embedding, Range of Embedding with Topology and Atlas Induced by Embedding Is Embedded Submanifold with Boundary of Codomain |
| description/proof of that for \(C^\infty\) embedding, range of embedding with topology and atlas induced by embedding is embedded submanifold with boundary of codomain |
| 853: Dihedral Group |
| definition of dihedral group |
| 854: Skewed Dihedral Group |
| definition of skewed dihedral group |
| 855: For Real or Complex Vectors Space with Inner Product, Linear Combination of Finite Vectors Cannot Be Perpendicular to Each Constituent Without Being 0 |
| description/proof of that for real or complex vectors space with inner product, linear combination of finite vectors cannot be perpendicular to each constituent without being 0 |
| 856: Wedge Sum of Pointed Sets |
| definition of wedge sum of pointed sets |
| 857: Wedge Sum of Pointed Topological Spaces |
| definition of wedge sum of pointed topological spaces |
| 858: Wedge Sum of Pointed Maps |
| definition of wedge sum of pointed maps |
| 859: For 2 Pointed Continuous Maps, Wedge Sum of Maps Is Continuous |
| description/proof of that for 2 pointed continuous maps, wedge sum of maps is continuous |
| 860: Identity Map from Subset of Euclidean \(C^\infty\) Manifold or Closed Upper Half Euclidean \(C^\infty\) Manifold with Boundary into Subset of Euclidean \(C^\infty\) Manifold or Closed Upper Half Euclidean \(C^\infty\) Manifold with Boundary Is \(C^\infty\) |
| description/proof of that identity map from subset of Euclidean \(C^\infty\) manifold or closed upper half Euclidean \(C^\infty\) manifold with Boundary into subset of Euclidean \(C^\infty\) manifold or closed upper half Euclidean \(C^\infty\) manifold with boundary is \(C^\infty\) |
| 861: Map from Open Subset of \(C^\infty\) Manifold with Boundary onto Open Subset of Euclidean \(C^\infty\) Manifold or Closed Upper Half Euclidean \(C^\infty\) Manifold with Boundary Is Chart Map iff It Is Diffeomorphism |
| description/proof of that map from open subset of \(C^\infty\) manifold with boundary onto open subset of Euclidean \(C^\infty\) manifold or closed upper half Euclidean \(C^\infty\) manifold with boundary is chart map iff it is diffeomorphism |
| 862: For Permutations Group, Its Element, and Element of Permutations Domain, Sequence of Power Operations of Element on Domain Element Returns Back from Domain Element |
| description/proof of that for permutations group, its element, and element of permutations domain, sequence of power operations of element on domain element returns back from domain element |
| 863: For Permutations Group, Its Element, Element of Permutations Domain, and Sequence of Power Operations of Element on Domain Element, Another Sequence with Another Domain Element Not Contained in 1st Sequence Is Disjoint from 1st Sequence |
| description/proof of that for permutations group, its element, element of permutations domain, and sequence of power operations of element on domain element, another sequence with another domain element not contained in 1st sequence is disjoint from 1st sequence |
| 864: For \(C^\infty\) Manifold with Boundary and Embedded Submanifold with Boundary, Around Each Point on Submanifold with Boundary, There Is Trivializing Open Subset for Manifold with Boundary Whose Intersection with Submanifold with Boundary Is Chart Domain |
| description/proof of that for \(C^\infty\) manifold with boundary and embedded submanifold with boundary, around each point on submanifold with boundary, there is trivializing open subset for manifold with boundary whose intersection with submanifold with boundary is chart domain |
| 865: For Set, To-Be-Atlas Determines Topology and Atlas |
| description/proof of that for set, to-be-atlas determines topology and atlas |
| 866: For Set and 2 Topology-Atlas Pairs, iff There Is Common Chart Domains Open Cover and Each Transition Is Diffeomorphism, Pairs Are Same |
| description/proof of that for set and 2 topology-atlas pairs, iff there is common chart domains open cover and each transition is diffeomorphism, pairs are same |
| 867: Restricted \(C^\infty\) Vectors Bundle |
| definition of restricted \(C^\infty\) vectors bundle |
| 868: Left or Right Coset of Subgroup by Element of Group |
| definition of left or right coset of subgroup by element of group |
| 869: Integers Modulo Natural Number Group |
| definition of integers modulo natural number group |
| 870: Field |
| definition of field |
| 871: Normalizer of Subgroup on Group |
| definition of normalizer of subgroup on group |
| 872: Sylow p-Subgroup of Group |
| definition of Sylow p-subgroup of group |
| 873: Restricted \(C^\infty\) Vectors Bundle W.r.t. Embedded Submanifold with Boundary Is Embedded Submanifold with Boundary |
| description/proof of that restricted \(c^\infty\) vectors bundle w.r.t. embedded submanifold with boundary is embedded submanifold with boundary |
| 874: For \(C^\infty\) Vectors Bundle and Section from Subset of Base Space \(C^k\) at Point Where \(0 \lt k\), There Is \(C^k\) Extension on Open-Neighborhood-of-Point Domain |
| description/proof of that for \(C^\infty\) vectors bundle and section from subset of base space \(C^k\) at point where \(0 \lt k\), there is \(C^k\) extension on open-neighborhood-of-point domain |
| 875: Integers Modulo Natural Number Ring |
| definition of integers modulo natural number ring |
| 876: Integers Modulo Prime Number Field |
| definition of integers modulo prime number field |
| 877: Quotient Ring of Commutative Ring by Ideal Is Commutative Ring |
| description/proof of that quotient ring of commutative ring by ideal is commutative ring |
| 878: Quotient Ring of Integers Ring by Prime Principal Ideal Is Field |
| description/proof of that quotient ring of integers ring by prime principal ideal is field |
| 879: For Topological Space, Union of Closures of Subsets Is Contained in Closure of Union of Subsets |
| description/proof of that for topological space, union of closures of subsets is contained in closure of union of subsets |
| 880: For Locally Finite Set of Subsets of Topological Space, Closure of Union of Subsets Is Union of Closures of Subsets |
| description/proof of that for locally finite set of subsets of topological space, closure of union of subsets is union of closures of subsets |
| 881: For \(C^\infty\) Vectors Bundle, \(C^\infty\) Section Along Closed Subset of Base Space Can Be Extended to Over Whole Base Space with Support Contained in Any Open Neighborhood of Subset |
| description/proof of that for \(C^\infty\) vectors bundle, \(C^\infty\) section along closed subset of base space can be extended to over whole base space with support contained in any open neighborhood of subset |
| 882: Set of \(C^\infty\) Sections of \(C^\infty\) Vectors Bundle Linearly Independent at Point Is Linearly Independent on Open Neighborhood of Point |
| description/proof of that set of \(C^\infty\) sections of \(C^\infty\) vectors bundle linearly independent at point is linearly independent on open neighborhood of point |
| 883: Algebra over Field |
| definition of algebra over field |
| 884: %Category Name% Automorphism |
| definition of %category name% automorphism |
| 885: For 2 Square Matrices over Commutative Ring, Trace of Product of Matrices Does Not Depend on Order of Product |
| description/proof of that for 2 square matrices over commutative ring, trace of product of matrices does not depend on order of product |
| 886: Trace of Vectors Space Endomorphism |
| definition of trace of vectors space endomorphism |
| 887: For \(C^\infty\) Manifold with Boundary and 2 Real Vectors Spaces, \(C^\infty\) Bijection from Product of Manifold with Boundary and Former Vectors Space into Product of Manifold with Boundary and Latter Vectors Space That Is 1st-Factor-Preserving and 1st-Factor-Fixed-Linear Is Diffeomorphism |
| description/proof of that for \(C^\infty\) manifold with boundary and 2 real vectors spaces, \(C^\infty\) bijection from product of manifold with boundary and former vectors space onto product of manifold with boundary and latter vectors space that is 1st-factor-preserving and 1st-factor-fixed-linear is diffeomorphism |
| 888: For 2 \(C^\infty\) Vectors Bundles over Same \(C^\infty\) Manifold with Boundary, Bijective \(C^\infy\) Vectors Bundle Homomorphism Is '\(C^\infty\) Vectors Bundles - \(C^\infty\) Vectors Bundle Homomorphisms' Isomorphism |
| description/proof of that for 2 \(C^\infty\) vectors bundles over same \(C^\infty\) manifold with boundary, bijective \(C^\infty\) vectors bundle homomorphism is '\(C^\infty\) vectors bundles - \(C^\infty\) vectors bundle homomorphisms' isomorphism |
| 889: Map Between \(C^\infty\) Manifolds with Boundary Is \(C^\k\) if and Only if Domain Restriction of Map to Each Element of Open Cover Is \(C^k\) |
| description/proof of that map between \(C^\infty\) manifolds with boundary is \(C^k\) if and only if domain restriction of map to each element of open cover is \(C^k\) |
| 890: Injective Map Between \(C^\infty\) Manifolds with Boundary Is \(C^\infty\) Embedding, if Domain Restriction of Map on Each Element of Open Cover Is \(C^\infty\) Embedding onto Open Subset of Range or Codomain |
| description/proof of that injective map between \(C^\infty\) manifolds with boundary is \(C^\infty\) embedding, if domain restriction of map on each element of open cover is \(C^\infty\) embedding onto open subset of range or codomain |
| 891: For \(C^\infty\) Vectors Bundle and \(C^\infty\) Local Frame over Open Subset, Around Each Point of Open Subset, There Is Possibly Smaller Chart for Bundle That Takes Components w.r.t. Frame |
| description/proof of that for \(C^\infty\) vectors bundle and \(C^\infty\) local frame over open subset, around each point of open subset, there is possibly smaller chart for bundle that takes components w.r.t. frame |
| 892: Composition of Map After Preimage Is Identical if Map Is Surjective w.r.t. Argument Subset |
| description/proof of that composition of map after preimage is identical if map is surjective w.r.t. argument subset |
| 893: For Map, Image of Subset Minus Subset Is Not Necessarily Image of 1st Subset Minus Image of 2nd Subset |
| description/proof of that for map, image of subset minus subset is not necessarily image of 1st subset minus image of 2nd subset |
| 894: For Map, Image of Subset Minus Subset Contains Image of 1st Subset Minus Image of 2nd Subset |
| description/proof of that for map, image of subset minus subset contains image of 1st subset minus image of 2nd subset |
| 895: For Injective Map, Image of Subset Minus Subset Is Image of 1st Subset Minus Image of 2nd Subset |
| description/proof of that for injective map, image of subset minus subset is image of 1st subset minus image of 2nd subset |
| 896: Map Preimage of Subset Minus Subset Is Preimage of 1st Subset Minus Preimage of 2nd Subset |
| description/proof of that map preimage of subset minus subset is preimage of 1st subset minus preimage of 2nd subset |
| 897: For Map, Subset of Domain, and Subset of Codomain, Image of Subset Is Contained in Subset and Image of Complement of Subset Is Contained in Complement of Subset, iff Preimage of Subset Is Subset and Preimage of Complement of Subset Is Complement of Subset |
| description/proof of that for map, subset of domain, and subset of codomain, image of subset is contained in subset and image of complement of subset is contained in complement of subset, iff preimage of subset is subset and preimage of complement of subset is complement of subset |
| 898: For Surjection, Preimages of Subsets Are Same iff Subsets Are Same |
| description/proof of that for surjection, preimages of subsets are same iff subsets are same |
| 899: On Set of Continuous Maps Between Topological Spaces, Being Homotopic Is Equivalence Relation |
| description/proof of that on set of continuous maps between topological spaces, being homotopic is equivalence relation |
| 900: hTop Category |
| definition of hTop category |
| 901: Homotopy Equivalence |
| definition of homotopy equivalence |
| 902: Adjunction Topological Space Is Hausdorff if Attaching-Destination Space Is Hausdorff, Attaching-Origin Space Is Regular, and Domain of Attaching-Map Is Closed and Retract of Open Neighborhood |
| description/proof of that adjunction topological space is Hausdorff if attaching-destination space is Hausdorff, attaching-origin space is regular, and domain of attaching-map is closed and retract of open neighborhood |
| 903: Retract of Hausdorff Topological Space Is Closed |
| description/proof of that retract of Hausdorff topological space is closed |
| 904: Intersection of Subsets Is Complement of Union of Complements of Subsets |
| description/proof of that intersection of subsets is complement of union of complements of subsets |
| 905: Union of Subsets Is Complement of Intersection of Complements of Subsets |
| description/proof of that union of subsets is complement of intersection of complements of subsets |
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