| Title |
|---|
| 1 Point Subset of Hausdorff Topological Space Is Closed |
| 1st-Countable Topological Space Is Sequentially Compact if It Is Countably Compact |
| 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 |
| 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 |
| 2 Continuous Maps into Hausdorff Topological Space That Disagree at Point Disagree on Neighborhood of Point |
| 2 Metrics with Condition with Each Other Define Same Topology |
| 2 Points Are Topologically Path-Connected iff There Is Path That Connects 2 Points |
| 2 Points That Are Path-Connected on Topological Subspace Are Path-Connected on Larger Subspace |
| 2 Points on Connected Lie Group Can Be Connected by Finite Left-Invariant Vectors Field Integral Curve Segments |
| 2 Points on Different Connected Components Are Not Path-Connected |
| 2 x 2 Special Orthogonal Matrix Can Be Expressed with Sine and Cosine of Angle |
| 2 x 2 Special Unitary Matrix Can Be Expressed with Sine and Cosine of Angle and Imaginary Exponentials of 2 Angles |
| Absolute Difference Between Complex Numbers Is or Above Difference Between Absolute Differences with Additional Complex Number |
| Accumulation Value of Net with Directed Index Set Is Convergence of Subnet |
| Affine Set Spanned by Non-Affine-Independent Set of Base Points on Real Vectors Space Is Affine Set Spanned by Affine-Independent Subset of Base Points |
| Area of Hyperrectangle Can Be Approximated by Area of Covering Finite Number Hypersquares to Any Precision |
| Area on Euclidean Metric Space Can Be Measured Using Only Hypersquares, Instead of Hyperrectangles |
| Basis Determines Topology |
| \(C^1\) Map from Open Set on Euclidean Normed \(C^\infty\) Manifold to Euclidean Normed \(C^\infty\) Manifold Locally Satisfies Lipschitz Condition |
| Canonical Map from Fundamental Group on Finite Product Topological Space into Product of Constituent Topological Space Fundamental Groups Is 'Groups - Group Homomorphisms' Isomorphism |
| Cantor Normal Form Is Unique |
| Cardinality of Multiple Times Multiplication of Set Is That Times Multiplication of Cardinality of Set |
| Categories Equivalence Is Equivalence Relation |
| Cauchy-Schwarz Inequality for Real or Complex Inner-Producted Vectors Space |
| Chain Rule for Derivative of Composition of \(C^1\), Euclidean-Normed Euclidean Vectors Spaces Maps |
| Characteristic Property of Disjoint Union |
| Characteristic Property of Product Topology |
| Characteristic Property of Subspace Topology |
| Chart on Regular Submanifold Is Extension of Adapting Chart |
| \(C^\infty\) Function on \(C^\infty\) Manifold Is \(C^\infty\) on Regular Submanifold |
| \(C^\infty\) Vectors Field Is Uniquely Defined by Its \(C^\infty\) Metric Value Functions with All \(C^\infty\) Vectors Fields |
| \(C^\infty\) Vectors Field on Regular Submanifold Is \(C^\infty\) as Vectors Field Along Regular Submanifold on Supermanifold |
| \(C^k\)-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 |
| Closed Continuous Map Between Topological Spaces with Compact Fibers Is Proper |
| Closed Discrete Subspace of Compact Topological Space Has Only Finite Points |
| Closed Set Minus Open Set Is Closed |
| Closed Set on Closed Topological Subspace Is Closed on Base Space |
| Closed Subspace of Locally Compact Topological Space Is Locally Compact |
| Closure of Difference of Subsets Is Not Necessarily Difference of Closures of Subsets, But Is Contained in Closure of Minuend |
| Closure of Normal Subgroup of Topological Group Is Normal Subgroup |
| Closure of Subgroup of Topological Group Is Subgroup |
| Closure of Subset Is Union of Subset and Accumulation Points Set of Subset |
| Closure of Union of Finite Subsets Is Union of Closures of Subsets |
| Collection of Sets That Are of Non-0 Cardinality Is Not Set |
| Compact Topological Space Has Accumulation Point of Subset with Infinite Points |
| Compactness of Topological Subset as Subset Equals Compactness as Subspace |
| Complement of Nowhere Dense Subset Is Dense |
| Complement of Open Dense Subset Is Nowhere Dense |
| Complement of Product of Subsets Is Union of Products of Whole Sets 1 of Which Is Replaced with Complement of Subset |
| Composition of Map After Preimage Is Contained in Argument Set |
| Composition of Map After Preimage Is Identical Iff Argument Set Is Subset of Map Image |
| Composition of Preimage After Map of Subset Contains Argument Set |
| Composition of Preimage After Map of Subset Is Identical If Map Is Injective with Respect to Argument Set Image |
| Composition of Preimage After Map of Subset Is Identical Iff It Is Contained in Argument Set |
| Compositions of Homotopic Maps Are Homotopic |
| Conjugation from Complex Numbers Euclidean Topological Space onto Complex Numbers Euclidean Topological Space Is Homeomorphism |
| Connected Component Is Closed |
| Connected Component Is Open on Locally Connected Topological Space |
| Connected Topological Component Is Exactly Connected Topological Subspace That Cannot Be Made Larger |
| Connected Topological Manifold Is Path-Connected |
| Connected Topological Subspaces of 1-Dimensional Euclidean Topological Space Are Intervals |
| Connection Depends Only on Section Values on Vector Curve |
| Continuous Embedding Between Topological Spaces with Closed Range Is Proper |
| Continuous Image of Path-Connected Subspace of Domain Is Path-Connected on Codomain |
| Continuous Map from Compact Topological Space into Hausdorff Topological Space Is Proper |
| Continuous Map from Topological Space into Hausdorff Topological Space with Continuous Left Inverse Is Proper |
| Continuous Surjection Between Topological Spaces Is Quotient Map if Any Codomain Subset Is Closed if Its Preimage Is Closed |
| Contraction Mapping Principle |
| Convex Set Spanned by Non-Affine-Independent Set of Base Points on Real Vectors Space Is Not Necessarily Affine Simplex Spanned by Affine-Independent Subset of Base Points |
| Convex Set Spanned by Possibly-Non-Affine-Independent Set of Base Points on Real Vectors Space Is Convex |
| Coordinates Matrix of Inverse Riemannian Metric Is Inverse of Coordinates Matrix of Riemannian Metric |
| Covering Map into Simply Connected Topological Space Is Homeomorphism |
| Criteria for Collection of Open Sets to Be Basis |
| Curves on Manifold as the \(C^\infty\) Right Actions of Curves That Represent Same Vector on Lie Group Represent Same Vector |
| Derivative of \(C^1\), Euclidean-Normed Euclidean Vectors Spaces Map Is Jacobian |
| Derived Operation of Monotone Continuous Operation from Ordinal Numbers Collection into Ordinal Numbers Collection Is Monotone Continuous |
| Descending Sequence of Ordinal Numbers Is Finite |
| Determinant of Square Matrix Whose Last Row Is All 1 and Whose Each Other Row Is All 0 Except Row Number + 1 Column 1 Is -1 to Power of Dimension + 1 |
| Difference of Map Images of Subsets Is Contained in Map Image of Difference of Subsets |
| Difference of Map Images of Subsets Is Map Image of Difference of Subsets if Map Is Injective |
| Disjoint Union of Closed Sets Is Closed in Disjoint Union Topology |
| Disjoint Union of Complements Is Disjoint Union of Whole Sets Minus Disjoint Union of Subsets |
| Double Dual of Finite Dimensional Real Vectors Space Is 'Vectors Spaces - Linear Morphisms' Isomorphic to Vectors Space |
| Dual of Finite Dimensional Real Vectors Space Constitutes Same Dimensional Vectors Space |
| Equivalence Between Derivation at Point of \(C^1\) Functions and Directional Derivative |
| Equivalence of Map Continuousness in Topological Sense and in Norm Sense for Coordinates Functions |
| Euclidean Topological Space Is 2nd Countable |
| Euclidean Topological Space Nested in Euclidean Topological Space Is Topological Subspace |
| Existence of Lie Group Neighborhood Whose Any Point Can Be Connected with Center by Left-Invariant Vectors Field Integral Curve |
| Expansion of Continuous Map on Codomain Is Continuous |
| Finite Dimensional Real Vectors Space Topology Defined Based on Coordinates Space Does Not Depend on Choice of Basis |
| Finite Dimensional Vectors Spaces Related by Linear Bijection Are of Same Dimension |
| Finite Intersection of Open Dense Subsets of Topological Space Is Open Dense |
| Finite Product of Compact Topological Spaces Is Compact |
| Finite Product of Locally Compact Topological Spaces Is Locally Compact |
| Finite Product of Sets Is Set |
| Finite Product of Topological Spaces Equals Sequential Products of Topological Spaces |
| Finite Union of Nowhere Dense Subsets of Topological Space Has Empty Interior |
| Fixed-Point in Proof of Veblen Fixed-Point Theorem Is Smallest That Satisfies Condition |
| For 1st Countable Topological Space, Some Facts About Points Sequences and Subset |
| 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 |
| 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 |
| For 2 Sets, Collection of Functions Between Sets Is Set |
| For Adjunction Topological Space, Canonical Map from Attaching-Destination Space to Adjunction Space Is Continuous Embedding |
| For Bijection, Preimage of Subset Under Inverse of Map Is Image of Subset Under Map |
| For \(C^\infty\) Function on Open Neighborhood, There Exists \(C^\infty\) Function on Manifold That Equals Function on Possibly Smaller Neighborhood |
| 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 |
| For \(C^\infty\) Map Between \(C^\infty\) Manifolds, Restriction of Map on Regular Submanifold Domain and Regular Submanifold Codomian Is \(C^\infty\) |
| For \(C^\infty\) Vectors Bundle, Global Connection Can Be Constructed with Local Connections over Open Cover, Using Partition of Unity Subordinate to Open Cover |
| For Compact \(C^\infty\) Manifold, Sequence of Points Has Convergent Subsequence |
| For Complete Metric Space, Closed Subspace Is Complete |
| For Coset Map with Respect to Subgroup, Preimage of Image of Subset Is Subgroup Multiplied by Subset |
| For Covering Map, 2 Lifts of Continuous Map from Connected Topological Space Totally Agree or Totally Disagree |
| For Covering Map, Criterion for Lift of Continuous Map from Path-Connected Locally Path-Connected Topological Space to Exist |
| For Covering Map, Lift of Product of Paths Is Product of Lifts of Paths |
| For Covering Map, Lift of Reverse of Path Is Reverse of Lift of Path |
| For Covering Map, There Is Unique Lift of Continuous Map from Finite Product of Closed Real Intervals for Each Initial Value |
| For Covering Map, There Is Unique Lift of Path for Each Point in Covering Map Preimage of Path Image of Point on Path Domain |
| 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 |
| For Disjoint Subset and Open Set, Closure of Subset and Open Set Are Disjoint |
| For Disjoint Union Topological Space, Inclusion from Constituent Topological Space to Disjoint Topological Space Is Continuous |
| 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 |
| For Euclidean Topological Space, Set of All Open Balls with Rational Centers and Rational Radii Is Basis |
| For Finite Set of Points on Real Vectors Space, if for Point, Set of Subtractions of Point from Other Points Is Linearly Independent, It Is So for Each Point |
| For Finite-Product Topological Space, Product of Neighborhoods Is Neighborhood |
| 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 |
| For Hausdorff Topological Space and 2 Disjoint Compact Subsets, There Are Disjoint Open Subsets Each of Which Contains Compact Subset |
| For Hausdorff Topological Space, Net with Directed Index Set Can Have Only 1 Convergence |
| 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 |
| For Infinite Product Topological Space and Closed Subset, Point on Product Space Whose Each Finite-Components-Projection Belongs to Corresponding Projection of Subset Belongs to Subset |
| For Infinite Product Topological Space and Subset, Point on Product Space Whose Each Finite-Components-Projection Belongs to Corresponding Projection of Subset Does Not Necessarily Belong to Subset |
| For Injective Closed Map Between Topological Spaces, Inverse of Codomain-Restricted-to-Range Map Is Continuous |
| 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 |
| 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 |
| 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 |
| For Locally Compact Hausdorff Topological Space, Around Point, There Is Open Neighborhood Whose Closure Is Compact |
| For Locally Compact Hausdorff Topological Space, in Neighborhood Around Point, There Is Open Neighborhood Whose Closure Is Compact and Contained in Neighborhood |
| For Locally Finite Cover of Topological Space, Compact Subset Intersects Only Finite Elements of Cover |
| For Locally Finite Open Cover of Topological Space, Closure of Union of Open Sets Is Union of Closures of Open Sets |
| 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 |
| 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 |
| 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 |
| 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 |
| 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 |
| 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 |
| 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 |
| 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 |
| For Map \(C^\infty\) at Point, Coordinates Function with Any Charts Is \(C^\infty\) at Point Image |
| 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 |
| For Maps Between Arbitrary Subsets of \(C^\infty\) Manifolds with Boundary \(C^k\) at Corresponding Points, Composition Is \(C^k\) at Point |
| 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 |
| For Maps Between Arbitrary Subsets of Euclidean \(C^\infty\) Manifolds \(C^k\) at Corresponding Points, Composition Is \(C^k\) at Point |
| For Metric Space, 1 Point Subset Is Closed |
| For Metric Space, Difference of Distances of 2 Points from Subset Is Equal to or Less Than Distance Between Points |
| 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 |
| For Monotone Continuous Operation from Ordinal Numbers Collection into Ordinal Numbers Collection, Image of Limit Ordinal Number Is Limit Ordinal Number |
| For Monotone Operation from Ordinal Numbers Collection into Ordinal Numbers Collection, Value Equals or Contains Argument |
| For Monotone Ordinal Numbers Operation, 2 Domain Elements Are in Membership Relation if Corresponding Images Are in Same Relation |
| 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 |
| For Normal Topological Space, Collapsed Topological Space by Closed Subset Is Normal |
| For Product Topological Space, Projection of Compact Subset Is Compact |
| For Product of 2 \(C^\infty\) Manifolds, Product for Which One of Constituents Is Replaced with Regular Submanifold Is Regular Submanifold |
| For Regular Topological Space, Collapsed Topological Space by Closed Subset Is Hausdorff |
| For Quotient Map, Codomain Subset Is Closed if Preimage of Subset Is Closed |
| For Quotient Map, Induced Map from Quotient Space of Domain by Map to Codomain Is Continuous |
| For Quotient Map, Its Restriction on Open or Closed Saturated Domain and on Restricted Image Codomain Is Quotient Map |
| For Regular Topological Space, Neighborhood of Point Contains Closed Neighborhood |
| For Sequence on Topological Space, Around Point, There Is Open Set That Contains Only Finite Points of Sequence if No Subsequence Converges to Point |
| For Set Plus Set as an Element, Open Sets That Are Subsets of Set and Subsets Whose Complements Are Finite Is Topology |
| For Set of Sequences for Fixed Domain and Codomain, Permutation Bijectively Maps Set onto Set |
| For Set of Sets, Dichotomically Nondisjoint Does Not Necessarily Mean Pair-Wise Nondisjoint |
| For Subset of Topological Space, Closure of Subset Minus Subset Has Empty Interior |
| For Topological Space and Point on Subspace, Intersection of Neighborhood of Point on Base Space and Subspace Is Neighborhood on Subspace |
| For Topological Space, Compact Subset of Subspace Is Compact on Base Space |
| For Topological Space, Intersection of Basis and Subspace Is Basis for Subspace |
| For Topological Space, Intersection of Compact Subset and Subspace Is Not Necessarily Compact on Subspace |
| For Topological Space, Open and Closed Subset of Space Is Union of Connected Components of Space |
| For Topological Space, Subset of Compact Subset Is Not Necessarily Compact |
| For Topological Space, Subspace Subset That Is Compact on Base Space Is Compact on Subspace |
| For Transfinite Recursion Theorem, Some Conditions with Which Partial Specifications of Formula Are Sufficient |
| For Transitive Set with Partial Ordering by Membership, Element Is Initial Segment Up to It |
| For Vectors Bundle, Chart Open Subset on Base Space Is Not Necessarily Trivializing Open Subset (Probably) |
| For Vectors Bundle, \(C^\infty\) Frame Exists Over and Only Over Trivializing Open Subset |
| For Vectors Bundle, Section over Trivializing Open Subset Is \(C^\infty\) iff Coefficients w.r.t. \(C^\infty\) Frame over There Are \(C^\infty\) |
| For Vectors Bundle, There Is Chart Trivializing Open Cover |
| For Vectors Bundle, Trivialization of Chart Trivializing Open Subset Induces Canonical Chart Map |
| 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 |
| For Well-Ordered Structure and Its Sub Structure, Ordinal Number of Sub Structure Is Member of or Is Ordinal Number of Base Structure |
| Formula That Uniquely Maps Each Element of Set into Set Constitutes Function |
| 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 |
| From Euclidean Normed Topological Space to Equal or Higher Dimension Euclidean Normed Topological Space Lipschitz Condition Satisfying Map Image of Measure 0 Subset Is Measure 0 |
| From Natural Number to Countable Set Functions Set Is Countable |
| Functionally Structured Topological Spaces Category Morphisms Are Morphisms |
| Fundamental Group Homomorphism Induced by Composition of Continuous Maps Is Composition of Fundamental Group Homomorphisms Induced by Maps |
| Fundamental Group Homomorphism Induced by Homeomorphism Is 'Groups - Group Homomorphisms' Isomorphism |
| Fundamental Group Homomorphism Induced by Homotopy Equivalence Is 'Groups - Group Homomorphisms' Isomorphism |
| Fundamental Theorem for Group Homomorphism |
| Fundamental Theorem of Calculus for Euclidean-Normed Spaces Map |
| Hausdorff Maximal Principle: Chain in Partially-Ordered Set Is Contained in Maximal Chain |
| Homeomorphic Topological Manifolds Can Have Equivalent Atlases |
| How Wedge Product as an Equivalence Class of Elements of Tensor Algebra Is Related with the Tensor Products Construct |
| Identity Map with Domain and Codomain Having Different Topologies Is Continuous iff Domain Is Finer than Codomain |
| If Union of Disjoint Subsets Is Closed, Each Subset Is Not Necessarily Closed |
| If Union of Disjoint Subsets Is Open, Each Subset Is Not Necessarily Open |
| Image of Continuous Map from Compact Topological Space to \(\mathbb{R}\) Euclidean Topological Space Has Minimum and Maximum |
| Inclusion into Topological Space from Closed Subspace Is Closed Continuous Embedding |
| Inclusion into Topological Space from Subspace Is Continuous |
| Induced Functional Structure on Continuous Topological Spaces Map Codomain Is Functional Structure |
| Induced Functional Structure on Topological Subspace by Inclusion Is Functional Structure |
| Induced Map from Domain Quotient of Continuous Map Is Continuous |
| 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 |
| Injective Map Image of Intersection of Sets Is Intersection of Map Images of Sets |
| Intersection of 2 Transversal Regular Submanifolds of \(C^\infty\) Manifold Is Regular Submanifold of Specific Codimension |
| Intersection of Closure of Subset and Open Subset Is Contained in Closure of Intersection of Subset and Open Subset |
| Intersection of Complements of Subsets Is Complement of Union of Subsets |
| Intersection of Products of Sets Is Product of Intersections of Sets |
| Intersection of Set of Transitive Relations Is Transitive |
| Intersection or Finite Union of Closed Sets Is Closed |
| Inverse Theorem for Euclidean-Normed Spaces Map |
| Inverse of Closed Bijection Is Continuous |
| Inverse of Partial Ordering Is Partial Ordering |
| In Nest of Topological Subspaces, Connected-ness of Subspace Does Not Depend on Superspace |
| In Nest of Topological Subspaces, Openness of Subset on Subspace Does Not Depend on Superspace |
| Tangent Vectors Space of General Linear Group of Finite-Dimensional Real Vectors Space at Identity Is 'Vectors Spaces - Linear Morphisms' Isomorphic to General Linear Lie Algebra |
| Left-Invariant Vectors Field on Lie Group Is \(C^\infty\) |
| Lifts, That Start at Same Point, of Path-Homotopic Paths Are Path-Homotopic |
| Limit Condition Can Be Substituted with With-Equal Conditions |
| Linear Image of Finite Dimensional Vectors Space Is Vectors Space |
| Linear Map Between Euclidean Topological Spaces Is Continuous |
| Linear Surjection from Finite Dimensional Vectors Space to Same Dimensional Vectors Space Is 'Vectors Spaces - Linear Morphisms' Isomorphism |
| Local Criterion for Openness |
| Local Characterization of Closure: Point Is on Closure of Subset iff Every Neighborhood of Point Intersects Subset |
| Local Unique Solution Existence for Euclidean-Normed Space ODE |
| Locally Compact Hausdorff Topological Space Is Paracompact iff Space Is Disjoint Union of Open \(\sigma\)-Compact Subspaces |
| Map Between Arbitrary Subsets of \(C^\infty\) Manifolds with Boundary Bijective and Locally Diffeomorphic at Each Point Is Diffeomorphism |
| Map Between Arbitrary Subsets of \(C^\infty\) Manifolds with Boundary Locally Diffeomorphic at Point Is \(C^\infty\) at Point |
| 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 Restricts to Original Map Whose Restricted Coordinates Function Is Continuous |
| Map Between Topological Spaces Is Continuous if Domain Restriction of Map to Each Closed Set of Finite Closed Cover is Continuous |
| Map Between Topological Spaces Is Continuous if Domain Restriction of Map to Each Open Set of Open Cover is Continuous |
| Map Between Topological Spaces Is Continuous iff Preimage of Each Closed Subset of Codomain Is Closed |
| 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 |
| Map Image of Intersection of Sets Is Contained in Intersection of Map Images of Sets |
| Map Image of Intersection of Sets Is Not Necessarily Intersection of Map Images of Sets |
| Map Image of Point Is On Subset Iff Point Is on Preimage of Subset |
| Map Image of Subset Is Contained in Subset iff Subset Is Contained in Preimage of Subset |
| Map Image of Union of Sets Is Union of Map Images of Sets |
| Map of Quotient Topology Is Quotient Map |
| Map Preimage of Intersection of Sets Is Intersection of Map Preimages of Sets |
| Map Preimage of Codomain Minus Set Is Domain Minus Preimage of Set |
| Map Preimage of Range Is Whole Domain |
| Map Preimage of Union of Sets Is Union of Map Preimages of Sets |
| Map Preimage of Whole Codomain Is Whole Domain |
| Map Preimages of Disjoint Subsets Are Disjoint |
| Map That Is Anywhere Locally Constant on Connected Topological Space Is Globally Constant |
| Map from Open Subset of \(C^\infty\) Manifold onto Open Subset of Euclidean \(C^\infty\) Manifold Is Chart Map iff It Is Diffeomorphism |
| Maps Composition Preimage Is Composition of Map Preimages in Reverse Order |
| Maximal Element of Set w.r.t. Inverse of Ordering Is Minimal Element of Set w.r.t. Original Ordering |
| Metric Space Is Compact iff Each Infinite Subset Has \(\omega\)-Accumulation Point |
| Minimal Element of Set w.r.t. Inverse of Ordering Is Maximal Element of Set w.r.t. Original Ordering |
| Minus Dedekind Cut Of Dedekind Cut Is Really Dedekind Cut |
| Multiplication of Matrix Made of Same Size Blocks by Matrix Made of Multiplicable Same Size Blocks Is Blocks-Wise |
| Multiplications of Cardinalities of Sets Are Associative |
| 'Natural Number'-th Power of Cardinality of Set Is That Times Multiplication of Cardinality |
| Net to Product Topological Space Converges to Point iff Each Projection After Net Converges to Component of Point |
| No 2 Sets Have Each Other as Members |
| No Set Has Itself as Member |
| Nonzero Multiplicative Translation from Complex Numbers Euclidean Topological Space onto Complex Numbers Euclidean Topological Space Is Homeomorphism |
| On 2nd-Countable Topological Space, Open Cover Has Countable Subcover |
| On \(T_1\) Topological Space, Point Is \(\omega\)-Accumulation Point of Subset iff It Is Accumulation Point of Subset |
| Open Sets Whose Complements Are Finite and Empty Set Is Topology |
| Open Set Complement of Measure 0 Subset Is Dense |
| Open Set Intersects Subset if It Intersects Closure of Subset |
| Open Set Minus Closed Set Is Open |
| Open Set on Euclidean Topological Space Has Rational Point |
| Open Set on Open Topological Subspace Is Open on Base Space |
| Open Subspace of Locally Compact Hausdorff Topological Space Is Locally Compact |
| Order of Powers |
| Ordinal Number Is Grounded and Its Rank Is Itself |
| Ordinal Number Is Limit Ordinal Number iff It Is Nonzero and Is Union of Its All Members |
| Pair of Open Sets of Connected Topological Space Is Finite-Open-Sets-Sequence-Connected |
| Pair of Elements of Open Cover of Connected Topological Space Is Finite-Open-Sets-Sequence-Connected Via Cover Elements |
| Parameterized Family of Vectors and Curve Induced by \(C^\infty\) Right Action of Lie Group Represent Same Vector If . . . |
| Part of Set Is Subset if There Is Formula That Determines Each Element of Set to Be in or out of Part |
| Path-Connected Topological Component Is Exactly Path-Connected Topological Subspace That Cannot Be Made Larger |
| Path-Connected Topological Component Is Open and Closed on Locally Path-Connected Topological Space |
| Point Is on Map Image of Subset if Preimage of Point Is Contained in Subset, but Not Only if |
| Point on Connected Lie Group Can Be Expressed as Finite Product of Exponential Maps |
| Preimage Under Surjection Is Saturated w.r.t. Surjection |
| Preimage by Product Map Is Product of Preimages by Component Maps |
| Preimage of Non-Zero Determinants of Matrix of Continuous Functions Is Open |
| Preimage Under Domain-Restricted Map Is Intersection of Preimage Under Original Map and Restricted Domain |
| Product Map of Continuous Maps Is Continuous |
| 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 |
| Product of Closed Sets Is Closed in Product Topology |
| Product of Connected Topological Spaces Is Connected |
| Product of Finite Number of Connected Topological Spaces Is Connected |
| Product of Path-Connected Topological Spaces Is Path-Connected |
| Product of Topological Subspaces Is Subspace of Product of Base Spaces |
| Products of Sets Are Associative in 'Sets - Map Morphisms' Isomorphism Sense |
| Projective Hyperplane Is Hausdorff |
| Pushforward Image of \(C^\infty\) Vectors Field Along Curve on Regular Submanifold into Supermanifold Under Inclusion Is \(C^\infty\) |
| Quotient Space of Compact Topological Space Is Compact |
| Quotient Topology Is Sole Finest Topology That Makes Map Continuous |
| Quotient of Cylinder with Antipodal Points Identified Is Homeomorphic to Möbius Band |
| Inner Product on Real or Complex Vectors Space Induces Norm |
| 'Real Vectors Spaces-Linear Morphisms' Isomorphism Between Topological Spaces with Coordinates Topologies Is Homeomorphic |
| Regular Submanifold of Regular Submanifold Is Regular Submanifold of Base \(C^\infty\) Manifold of Specific Codimension |
| Relation Between Power Set Axiom and Subset Axiom |
| Residue of Derivative of Normed-Spaces Map Is Differentiable at Point If ..., and the Derivative Is ... |
| Restriction of \(C^\infty\) Map on Open Domain and Open Codomain Is \(C^\infty\) |
| Restriction of \(C^\infty\) Vectors Bundle on Regular Submanifold Base Space Is \(C^\infty\) Vectors Bundle |
| Restriction of Continuous Embedding on Domain and Codomain Is Continuous Embedding |
| Restriction of Continuous Map on Domain and Codomain Is Continuous |
| Restriction of Proper Map Between Topological Spaces on Saturated Domain Subset and Range Codomain Is Proper |
| Reverse of Tietze Extension Theorem |
| Riemannian Bundle Has Compatible Connection |
| Rotation in \(n\)-Dimensional Euclidean Vectors Space Is Same \(2\)-Dimensional Rotations Along \((n - 2)\)-Dimensional Subspace Axis |
| Set of Neighborhood Bases at All Points Determines Topology |
| Set of Subsets Around Each Point with Conditions Generates Unique Topology with Each Set Being Neighborhood Basis |
| Set of Subsets with Whole Set and Empty Set Constitutes Subbasis |
| Set of Vectors Space Homomorphisms Constitutes Vectors Space |
| Set of n x n Quaternion Matrices Is 'Rings - Homomorphism Morphisms' Isomorphic to Set of Corresponding 2n x 2n Complex Matrices |
| Simplex Is Homeomorphic to Same-Dimensional Closed Ball |
| Some Facts about Separating Possibly-Higher-than-2-Dimensional Matrix into Symmetric Part and Antisymmetric Part w.r.t. Indices Pair |
| Some Para-Product Maps of Continuous Maps Are Continuous |
| Some Parts of Legitimate Formulas for ZFC Set Theory |
| Some Properties Concerning Adjunction Topological Space When Inclusion to Attaching-Origin Space from Subset Is Closed Embedding |
| Square of Euclidean Norm of \(\mathbb{R}^n\) Vector Is Equal to or Larger Than Positive Definite Real Quadratic Form Divided by Largest Eigenvalue |
| Square of Euclidean Norm of \(\mathbb{R}^n\) Vector Is Equal to or Smaller Than Positive Definite Real Quadratic Form Divided by Smallest Eigenvalue |
| Stereographic Projection Is Homeomorphism |
| Subgroup of Abelian Additive Group Is Retract of Group Iff There Is Another Subgroup Such That Group is Sum of Subgroups |
| Subset Is Contained in Map Preimage of Image of Subset |
| Subset Minus Subset Is Complement of 2nd Subset Minus Complement of 1st Subset |
| Subset Minus Union of Sequence of Subsets Is Intersection of Subsets Each of Which Is 1st Subset Minus Partial Union of Sequence |
| Subset of 1st Category Subset Is of 1st Category |
| Subset of Non-Open Topological Subspace Is Open on Subspace If It Is Open on Base Space |
| Subset of Open Topological Subspace Is Open on Subspace Iff It Is Open on Base Space |
| 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 |
| Subset of Quotient Topological Space Is Closed iff Preimage of Subset Under Quotient Map Is Closed |
| Subset of \(R^{d-k}\) Is Open If the Product of \(R^k\) and Subset Is Open |
| Subset of Subspace of Adjunction Topological Space Is Open Iff Projections of Preimage of Subset Are Open with Condition |
| Subset on Topological Subspace Is Closed iff There Is Closed Set on Base Space Whose Intersection with Subspace Is Subset |
| Subspace That Contains Connected Subspace and Is Contained in Closure of Connected Subspace Is Connected |
| Subspace of 2nd Countable Topological Space Is 2nd Countable |
| Sufficient Conditions for Existence of Unique Global Solution on Interval for Euclidean-Normed Euclidean Vectors Space ODE |
| Superset of Residual Subset Is Residual |
| There Are Rational and Irrational Dedekind Cuts Between 2 Dedekind Cuts |
| There Is No Set That Contains All Sets |
| Topological Connected-ness of 2 Points Is Equivalence Relation |
| Topological Path-Connected-ness of 2 Points Is Equivalence Relation |
| 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 |
| Topological Space Is Connected if Quotient Space and Each Element of Quotient Space Are Connected |
| Topological Space Is Connected iff Its Open and Closed Subsets Are Only It and Empty Set |
| Topological Space Is Countably Compact if It Is Sequentially Compact |
| Topological Space Is Countably Compact iff Each Infinite Subset Has \(\omega\)-Accumulation Point |
| Topological Space Is Normal Iff for Closed Set and Its Containing Open Set There Is Closed-Set-Containing Open Set Whose ~ |
| Topological Subspace Is Locally Closed iff It Is Intersection of Closed Subset and Open Subset of Base Space |
| Topological Subspaces Map Continuousness at Point Is Implied by Continuousness of Map 'Open Set'-Wise Extended to Superspaces |
| Topological Sum of Paracompact Topological Spaces Is Paracompact |
| Transitive Closure of Subset Is Transitive Set That Contains Subset |
| Unbounded Collection of Ordinal Numbers Is Not Set |
| Union of 2 Connected Subspaces Is Connected if Each Neighborhood of Point on Subspace Contains Point of Other Subspace |
| Union of Complements of Subsets Is Complement of Intersection of Subsets |
| Union of Dichotomically Nondisjoint Set of Real Intervals Is Real Interval |
| Union of Indexed Subsets Minus Union of Subsets Indexed with Same Indices Set Is Contained in Union of Subset Minus Subset for Each Index |
| Union of Path-Connected Subspaces Is Path-Connected if Subspace of Point from Each Subspace Is Path-Connected |
| Unique Existence of Monoid Identity Element |
| Universal Property of Continuous Embedding |
| Universal Property of Quotient Map |
| Vectors Field Along \(C^\infty\) Curve Is \(C^\infty\) iff Operation Result on Any \(C^\infty\) Function is \(C^\infty\) |
| Vectors Field Is \(C^\infty\) If and Only If Operation Result on Any \(C^\infty\) Function Is \(C^\infty\) |
| 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 |
| Velocity Vectors Field Along \(C^\infty\) Curve Is \(C^\infty\) |
| Well-Ordered Subset with Inclusion Ordering Is Chain in Base Set |
| What Chart Induced Basis Vector on \(C^\infty\) Manifold with Boundary Is |
| What Velocity of Curve at Closed Boundary Point Is |
| When Image of Point Is on Image of Subset, Point Is on Subset if Map Is Injective with Respect to Image of Subset |
| Why Local Solution Existence Does Not Guarantee Global Existence for Euclidean-Normed Space ODE |
| With Respect to Normal Subgroup, Set of Cosets Forms Group |
| With Respect to Subgroup, Coset by Element of Group Equals Coset iff Element Is Member of Latter Coset |
| 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 |
| n-Sphere Is Path-Connected |
