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 |
A definition of locally trivial surjection of rank \(r\) |
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 |
A definition of \(C^\infty\) vectors bundle |
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 |
A 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 |
A 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 |
A 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 |
A definition of metric space |
31: Lie Algebra |
A 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 there is an isomorphism between tangent space of general linear group at identity and general linear Lie algebra |
38: Derivative of Real-1-Parameter Family of Vectors |
A definition of derivative of real-1-parameter family of vectors |
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 |
A description/proof of that connected topological manifold is path-connected |
48: Path Can Be Finitely Open-Covered |
A description/proof of that path can be finitely open-covered |
49: Limit Condition Can Be Substituted with With-Equal Conditions |
A description/proof of that limit condition 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 Manifold That Equals Function on Possibly Smaller Neighborhood |
A description/proof of that for \(C^\infty\) function on open neighborhood, there exists \(C^\infty\) function on manifold 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 |
A definition of structure |
80: %Structure Kind Name% Homomorphism |
A definition of %structure kind name% homomorphism |
81: Category |
A definition of category |
82: Morphism |
A definition of morphism |
83: Covariant Functor |
A definition of covariant functor |
84: Contravariant Functor |
A 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 |
A definition of ideal of ring |
91: Quotient Ring of Ring |
A definition of quotient ring of ring |
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 |
A description/proof of that 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 |
A 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 |
A description/proof of that union of complements of subsets is complement of intersection of subsets |
101: Subspace Topology |
A definition of subspace topology |
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 Non-Open Topological Subspace Is Open on Subspace If It Is Open on Base Space |
A description/proof of that subset of non-open topological subspace is open on subspace if it is open on base space |
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 |
A 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 Image of Finite Dimensional Vectors Space Is Vectors Space |
A description/proof of that linear image of finite dimensional vectors space is vectors space |
118: %Category Name% Isomorphism |
A 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: Topological Subspaces Map Continuousness at Point Is Implied by Continuousness of Map 'Open Set'-Wise Extended to Superspaces |
A description/proof of that topological subspaces map continuousness at point is implied by continuousness of map 'open set'-wise extended to superspaces |
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 |
A definition of convergence of net with directed indices 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 |
A 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 |
A description/proof of that composition of map after preimage is identical iff argument set is subset of map image |
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 |
A description/proof of that local characterization of closure: point is on closure of subset iff its every neighborhood 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 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 |
A 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 |
A 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 |
A description/proof of that 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 |
408: For Vectors Bundle, There Is Chart Trivializing Open Cover |
A description/proof of that for vectors bundle, there is chart trivializing open cover |
409: For Vectors Bundle, Trivialization of Chart Trivializing Open Subset Induces Canonical Chart Map |
A description/proof of that for 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 |
A description/proof of that for 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) |
A description/proof of that for 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 |
A description/proof of that for 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 |
A 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 |
A 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 |
A 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 |
A 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 |
A 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 |
A 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 |
A 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 |
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