2023-02-05

0: The Table of Contents of the Series, 'Definitions and Propositions'

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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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