| 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 |
| 6-Elements Group Cannot Have 2 3-Elements Subgroups That Share Only Identity |
| Abelian Group Is Simple Group iff Its Order Is Prime Number |
| 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 |
| Adjunction Topological Space Is Hausdorff if Attaching-Destination Space Is Hausdorff, Attaching-Origin Space Is Regular, and Domain of Attaching-Map Is Closed and Retract of Open Neighborhood |
| Affine Map from Affine or Convex Set Spanned by Possibly-Non-Affine-Independent Set of Base Points on Finite-Dimensional Real Vectors Space into Finite Dimensional Real Vectors Space Is Continuous W.r.t. Canonical Topologies |
| Affine Map from Affine or Convex Set Spanned by Possibly-Non-Affine-Independent Set of Base Points on Real Vectors Space Is Linear |
| 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 |
| Affine Simplex Map into Finite-Dimensional Vectors Space Is Continuous w.r.t. Canonical Topologies |
| Affine Simplex on Finite-Dimensional Real Vectors Space Is Closed and Compact on Canonical Topological Superspace |
| Affine Subset of Finite-Dimensional Real Vectors Space Is Spanned by Finite Affine-Independent Set of Base Points |
| Antisymmetrized After Symmetrized Same-Length Multi-Dimensional Array or Symmetrized After Antisymmetrized Same-Length Multi-Dimensional Array Is 0 |
| 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 |
| Bijective Group Homomorphism Is 'Groups - Homomorphisms' Isomorphism |
| Bijective Lie Algebra Homomorphism Is 'Lie Algebras - Homomorphisms' Isomorphism |
| Bijective Linear Map Is 'Vectors Spaces - Linear Morphisms' Isomorphism |
| Boundary of Subset of Topological Space Is Set of Points of Each of Which Each Neighborhood Intersects Both Subset and Complement of Subset |
| \(C^1\) Map from Open Set on Euclidean Normed \(C^\infty\) Manifold to Euclidean Normed \(C^\infty\) Manifold Locally Satisfies Lipschitz Condition |
| Cancellation Rule on Integral Domain |
| 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 |
| Closed Upper Half Euclidean Topological Space Is Homeomorphic to Product of Lower-Dimensional Euclidean Spaces and Closed Upper Half Euclidean Space |
| 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 Affine Maps Is Affine Map |
| Composition of \(C^\infty\) Embedding After Diffeomorphism or Diffeomorphism After \(C^\infty\) Embedding Is \(C^\infty\) Embedding |
| Composition of Map After Preimage Is Contained in Argument Set |
| Composition of Map After Preimage Is Identical if Map Is Surjective w.r.t. Argument Subset |
| 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 |
| Composition of Product Maps Is Product of Compositions of Component Maps |
| 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 |
| Contraction of 2 Same-Length Multi-Dimensional Arrays One of Which Is Symmetrized or Antisymmetrized w.r.t. Set of Indexes Is Contraction with Also Other Array Symmetrized or Antisymmetrized Accordingly |
| 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 |
| Domain of Affine Simplex Map Is Closed and Compact on Euclidean Topological Superspace |
| 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 |
| Element of Simplicial Complex on Finite-Dimensional Real Vectors Space Is Closed and Compact on Underlying Space of Complex |
| 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 Is Homeomorphic to Product of Lower-Dimensional Euclidean Spaces |
| 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 Embedding on Codomain Is Continuous Embedding |
| Expansion of Continuous Map on Codomain Is Continuous |
| Field Is Integral Domain |
| Finite Composition of Bijections Is Bijection, if Codomains of Constituent Bijections Equal Domains of Succeeding Bijections |
| Finite Composition of Injections Is Injection |
| Finite Composition of Motions Is Motion |
| Finite Composition of Surjections Is Not Necessarily Surjection |
| Finite Composition of Surjections Is Surjection, if Codomains of Constituent Surjections Equal Domains of Succeeding Surjections |
| 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 Direct Product of Groups Is 'Groups - Homomorphisms' Isomorphic to Direct Product of Corresponding Isomorphic Groups |
| Finite Intersection of Open Dense Subsets of Topological Space Is Open Dense |
| Finite Product of 2nd-Countable Topological Spaces Is 2nd-Countable |
| Finite Product of Compact Topological Spaces Is Compact |
| Finite Product of Locally Compact Topological Spaces Is Locally Compact |
| Finite Product of Normal Subgroups Is Commutative and Is Normal Subgroup |
| Finite Product of Sets Is Set |
| Finite Product of Subgroups Is Associative |
| 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 \(C^\infty\) Vectors Bundles over Same \(C^\infty\) Manifold with Boundary, Bijective \(C^\infty\) Vectors Bundle Homomorphism Is '\(C^\infty\) Vectors Bundles - \(C^\infty\) Vectors Bundle Homomorphisms' Isomorphism |
| For 2 Decompositions of Vector with Common Constituent, Coefficients of Common Constituent Are Same if Common Constituent Is Not on Vectors Space Spanned by Other Constituents |
| 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 Pointed Continuous Maps, Wedge Sum of Maps Is Continuous |
| For 2 Sets, Collection of Functions Between Sets Is Set |
| For 2 Sets, Collection of Relations Between Sets Is Set |
| For 2 Square Matrices over Commutative Ring, Trace of Product of Matrices Does Not Depend on Order of Product |
| For 2 Vectors Spaces That Share Operations on Intersection, Intersection Is Vectors Space |
| For Adjunction Topological Space, Canonical Map from Attaching-Destination Space to Adjunction Space Is Continuous Embedding |
| For Affine Simplex and Ascending Sequence of Faces, Set of Barycenters of Faces Is Affine-Independent |
| For Affine Simplex, Ascending Sequence of Faces, and Set of Barycenters of Faces, Convex Combination of Subset of Set of Barycenters Is Convex Combination W.r.t. Set of Vertexes of Affine Simplex |
| For Affine Simplex, Simplex Interior, and Vertex, Line Segment from Point on Simplex Interior to Vertex Is Contained in Union of Simplex Interior and Vertex |
| For Bijection, Preimage of Subset Under Inverse of Map Is Image of Subset Under Map |
| For \(C^\infty\) Embedding Between \(C^\infty\) Manifolds with Boundary, Restriction of Embedding on Embedded Submanifold with Boundary Domain Is \(C^\infty\) Embedding |
| For \(C^\infty\) Embedding, Range of Embedding with Topology and Atlas Induced by Embedding Is Embedded Submanifold with Boundary of Codomain |
| For \(C^\infty\) Function on Open Neighborhood, There Exists \(C^\infty\) Function on \(C^\infty\) Manifold with Boundary That Equals Function on Possibly Smaller Neighborhood |
| For \(C^\infty\) Immersion Between \(C^\infty\) Manifolds with Boundary, Its Global Differential Is \(C^\infty\) Immersion |
| For \(C^\infty\) Manifold with Boundary and 2 Real Vectors Spaces, \(C^\infty\) Bijection from Product of Manifold with Boundary and Former Vectors Space into Product of Manifold with Boundary and Latter Vectors Space That Is 1st-Factor-Preserving and 1st-Factor-Fixed-Linear Is Diffeomorphism |
| For \(C^\infty\) Manifold with Boundary and Chart, Restriction of Chart on Open Subset Domain Is Chart |
| For \(C^\infty\) Manifold with Boundary and Embedded Submanifold with Boundary, Around Each Point on Submanifold with Boundary, There Is Trivializing Open Subset for Manifold with Boundary Whose Intersection with Submanifold with Boundary Is Chart Domain |
| For \(C^\infty\) Manifold with Boundary and Embedded Submanifold with Boundary, Inverse of Codomain Restricted Inclusion Is \(C^\infty\) |
| 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 \(C^\infty\) Manifold with Boundary and Open Submanifold with Boundary, Differential of Inclusion at Point on Open Submanifold with Boundary Is 'Vectors Spaces- Linear Morphisms' Isomorphism |
| For \(C^\infty\) Manifold with Boundary and Regular Domain, Differential of Inclusion at Point on Regular Domain Is 'Vectors Spaces - Linear Morphisms' Isomorphism |
| For \(C^\infty\) Manifold with Boundary, Embedded Submanifold with Boundary of Embedded Submanifold with Boundary Is Embedded Submanifold with Boundary of Manifold with Boundary |
| For \(C^\infty\) Manifold with Boundary, Interior Point Has Chart Ball and Boundary Point Has Chart Half Ball |
| For \(C^\infty\) Manifold with Boundary, Interior Point Has Chart Whose Range Is Whole Euclidean Space and Boundary Point Has Chart Whose Range Is Whole Half Euclidean Space |
| For \(C^\infty\) Manifold, Embedded Submanifold with Boundary, and \(C^\infty\) Vectors Field over Submanifold with Boundary, Differential by Inclusion After Vectors Field Is \(C^\infty\) over Submanifold with Boundary |
| For \(C^\infty\) Manifold, Regular Domain, \(C^\infty\) Manifold with Boundary, and \(C^\infty\) Map from Regular Domain into \(C^\infty\) Manifold with Boundary, Corresponding Map with Domain Regarded as Subset of Manifold Is \(C^\infty\) |
| For \(C^\infty\) Manifold, Subset, and Point on Subset, if Chart Satisfies Local Slice Condition for Embedded Submanifold or Local Slice Condition for Embedded Submanifold with Boundary, Its Sub-Open-Neighborhood Does So |
| For \(C^\infty\) Map Between \(C^\infty\) Manifolds, Restriction of Map on Regular Submanifold Domain and Regular Submanifold Codomian Is \(C^\infty\) |
| For \(C^\infty\) Map from Finite-Product \(C^\infty\) Manifold with Boundary, Induced Map with Set of Components of Domain Fixed Is \(C^\infty\) |
| For \(C^\infty\) Vectors Bundle and \(C^\infty\) Local Frame over Open Subset, Around Each Point of Open Subset, There Is Possibly Smaller Chart for Bundle That Takes Components w.r.t. Frame |
| For \(C^\infty\) Vectors Bundle and Section from Subset of Base Space \(C^k\) at Point Where \(0 \lt k\), There Is \(C^k\) Extension on Open-Neighborhood-of-Point Domain |
| For \(C^\infty\) Vectors Bundle, \(C^\infty\) Section Along Closed Subset of Base Space Can Be Extended to Over Whole Base Space with Support Contained in Any Open Neighborhood of Subset |
| 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 Commutative Ring, if Each Elements Pair Has Greatest Common Divisor, Each Finite Subset Has Greatest Common Divisor, Which Can Be Gotten Sequentially |
| For Compact \(C^\infty\) Manifold, Sequence of Points Has Convergent Subsequence |
| For Complete Metric Space, Closed Subspace Is Complete |
| For Continuous Map from Product Topological Space into Topological Space, Induced Map with Set of Components of Domain Fixed Is Continuous |
| 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, Cardinalities of Sheets Are Same |
| 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 \(C^\infty\) Manifold, Open Ball Is Diffeomorphic to Whole Space |
| For Euclidean Topological Space, Lower-Dimensional Euclidean Topological Space, Slicing Map, Projection, and Inclusion, Inclusion after Projection after Slicing Map Equals Slicing Map, and Projection after Slicing Map of Open Neighborhood of Point Is Open Neighborhood of Projection of Point |
| 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 Simplicial Complex on Finite-Dimensional Real Vectors Space, Simplex Interior of Maximal Simplex Is Open on Underlying Space of Complex |
| For Finite Simplicial Complex, Stars of Vertexes of Simplexes Is Open Cover of Underlying Space |
| For Finite \(p\)-Group, for Natural Number Smaller Than Power to Which \(p\) Is Order of Group, There Is Normal Subgroup of Group Whose Order Is \(p\) to Power of Natural Number |
| For Finite-Dimensional Normed Real Vectors Space with Canonical Topology, Norm Map Is Continuous |
| For Finite-Dimensional Vectors Space Basis, Replacing Element by Linear Combination of Elements with Nonzero Coefficient for Element Forms Basis |
| For Finite-Dimensional Vectors Space and Basis, Linearly Independent Set of Elements Can Be Augmented with Some Elements of Basis to Be Basis |
| For Finite-Dimensional Vectors Space and Basis, Vectors Space Is 'Vectors Spaces - Linear Morphisms' Isomorphic to Components Vectors Space |
| For Finite-Dimensional Vectors Space, Linearly Independent Subset Can Be Expanded to Be Basis by Adding Finite Elements |
| For Finite-Dimensional Vectors Space, Linearly Independent Subset with Dimension Number of Elements Is Basis |
| For Finite-Dimensional Vectors Space, Proper Subspace Has Lower Dimension |
| For Finite-Dimensional Vectors Space, Subset That Spans Space Can Be Reduced to Be Basis |
| For Finite-Dimensional Vectors Space, There Is No Basis That Has More Than Dimension Elements |
| For Finite-Dimensional Vectors Space, There Is No Linearly Independent Subset That Has More Than Dimension Elements |
| For Finite-Product Topological Space, Product of Neighborhoods Is Neighborhood |
| For Group Action, Induced Map with Fixed Group Element Is Bijection |
| For Group and Element, if There Is Positive Natural Number to Power of Which Element Is 1 and There Is No Smaller Such, Subgroup Generated by Element Consists of Element to Non-Negative Powers Smaller Than Number |
| For Group and Element, if There Is Positive Natural Number to Power of Which Element Is 1 and There Is No Smaller Such, Integers of Which Powers to Which Element Are 1 Are Only Multiples of Number |
| For Group and Finite-Order Element, Conjugate of Element Has Order of Element |
| For Group and Finite-Order Element, Inverse of Element Has Order of Element |
| For Group and Finite-Order Element, Order Power of Element Is \(1\) and Subgroup Generated by Element Consists of Element to Non-Negative Powers Smaller Than Element Order |
| For Group and Normal Subgroup, if Normal Subgroup and Quotient of Group by Normal Subgroup Are p-Groups, Group Is p-Group |
| For Group and Subgroup, Conjugation for Subgroup by Group Element Is 'Groups Homomorphisms' Isomorphism |
| For Group, Conjugation by Element Is 'Groups - Homomorphisms' Isomorphism |
| For Group, Normal Subgroup, and Quotient Group, Representatives Set Multiplied by Element Is Representatives Set |
| For Group, Normal Subgroup, and Subgroup, Subsets of Quotient Group That Contain Cosets of Subgroup Are Same or Disjoint |
| For Group, Multiplication Map with Fixed Element from Left or Right Is Bijection |
| For Group, Powers Sequence of Element That Returns Back Returns to Element |
| For Group, Subgroup, and Element of Group, if \(k\) Is 1st Positive Power to Which Element Belongs to Subgroup, Multiples of \(k\) Are Only Powers to Which Element Belongs to Subgroup |
| 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 Group and Its Subgroup, Subgroup Is Normal Subgroup if Its Conjugate Subgroup by Each Element of Group Is Contained in It |
| For Group as Direct Sum of Finite Number of Normal Subgroups, Element Is Uniquely Decomposed and Decomposition Is Commutative |
| For Group as Direct Sum of Finite Number of Normal Subgroups, Product of Subset of Normal Subgroups Is Group as Direct Sum of Subset |
| For Half Euclidean \(C^\infty\) Manifold with Boundary, Open Half Ball Is Diffeomorphic to Whole Space |
| 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 Map, Image of Subset Minus Subset Is Image of 1st Subset Minus Image of 2nd Subset |
| 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 Integral Domain, if Greatest Common Divisors of Subset Exist, They Are Associates of a Greatest Common Divisor |
| For Integral Domain, if Least Common Multiples of Subset Exist, They Are Associates of a Least Common Multiple |
| For Integral Domain, if Principal Ideal by Element Is Also by Another Element, Elements Are Associates with Each Other, and Principal Ideal Is by Any Associate |
| 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 Invertible Square Matrix, from Top Row Downward Through Any Row, Each Row Can Be Changed to Have 1 1 Component and 0 Others Without Duplication to Keep Matrix Invertible |
| For Linear Map from Finite-Dimensional Vectors Space, There Is Domain Subspace That Is 'Vectors Spaces - Linear Morphisms' Isomorphic to Range by Restriction of Map |
| For Linear Surjection Between Finite-Dimensional Vectors Spaces, Dimension of Codomain Is Equal to or Smaller than That of Domain |
| For Linear Surjection from Finite-Dimensional Vectors Space, if Dimension of Codomain Is Equal to or Larger than That of Domain, Surjection Is Bijection |
| For Linearly Independent Finite Subset of Module, Induced Subset of Module with Some Linear Combinations Is Linearly Independent |
| For Linearly Independent Sequence in Vectors Space, Derived Sequence in Which Each Element Is Linear Combination of Equal or Smaller Index Elements with Nonzero Equal Index Coefficient Is Linearly Independent |
| 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 Locally Finite Set of Subsets of Topological Space, Closure of Union of Subsets Is Union of Closures of Subsets |
| 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 Maps Between Arbitrary Subspaces of Topological Spaces Continuous at Corresponding Points, Composition Is Continuous at Point |
| For Map Between Embedded Submanifolds with Boundary of \(C^\infty\) Manifolds with Boundary, \(C^k\)-ness Does Not Change When Domain or Codomain Is Regarded to Be Subset |
| For Map Between Measurable Spaces, if Preimage of Each Element of Generator of Codomain \(\sigma\)-Algebra Is Measurable, Map Is Measurable |
| For Map Between Normed Vectors Spaces s.t. Image Norm Divided by Argument Norm Converges to 0 When Argument Norm Nears 0, Image Norm of Map Plus Nonzero Linear Map Divided by Argument Norm Does Not Do So |
| 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 Between Topological Spaces and Domain Point, if There Are Superspaces of Domain and Codomain, Open Neighborhoods of Point and of Point Image on Superspaces, and Continuous Map from Domain Neighborhood into Codomain neighborhood That Is Restricted to Original Map on Intersection of Domain Neighborhood and Original Domain, Original Map Is Continuous at Point |
| For Map \(C^\infty\) at Point, Coordinates Function with Any Charts Is \(C^\infty\) at Point Image |
| For Map from Subset of \(C^\infty\) Manifold with Boundary into Subset of \(C^\infty\) Manifold \(C^k\) at Point, There Is \(C^k\) Extension on Open-Neighborhood-of-Point Domain |
| For Map from Subset of \(C^\infty\) Manifold with Boundary into Subset of \(C^\infty\) Manifold with Boundary, Map Is Local Diffeomorphism iff for Each Domain Point and Its Image, There Are Charts by Which Coordinates Function Is Diffeomorphism |
| 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 Map, Image of Subset Minus Subset Contains Image of 1st Subset Minus Image of 2nd Subset |
| For Map, Image of Subset Minus Subset Is Not Necessarily Image of 1st Subset Minus Image of 2nd Subset |
| For Map, Subset of Domain, and Subset of Codomain, Image of Subset Is Contained in Subset and Image of Complement of Subset Is Contained in Complement of Subset, iff Preimage of Subset Is Subset and Preimage of Complement of Subset Is Complement of Subset |
| For Map, if There Is Inverse Direction Map Which After Original Map Is Identity, Original Map Is Injective |
| 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 Motion Between Real Vectors Spaces with Norms Induced by Inner Products That Fixes 0, Orthonormal Subset of Domain Is Mapped to Orthonormal Subset |
| For Motion Between Same-Finite-Dimensional Real Vectors Spaces with Norms Induced by Inner Products That Fixes 0, Motion Is Orthogonal Linear Map |
| For Motion Between Same-Finite-Dimensional Real Vectors Spaces with Norms Induced by Inner Products, Motion Is Bijective |
| 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 Nonzero Linear Map Between Normed Vectors Spaces, Image Norm Divided by Argument Norm Does Not Converge to 0 When Argument Norm Nears 0 |
| For Normal Topological Space, Collapsed Topological Space by Closed Subset Is Normal |
| For Open Subset of \(d_1\)-Dimensional Euclidean \(C^\infty\) Manifold, \(C^\infty\) Map into \(d_2\)-Dimensional Euclidean \(C^\infty\) Manifold Divided by Never-Zero \(C^\infty\) Map into 1-Dimensional Euclidean \(C^\infty\) Manifold Is \(C^\infty\) |
| For Permutations Group, Its Element, Element of Permutations Domain, and Sequence of Power Operations of Element on Domain Element, Another Sequence with Another Domain Element Not Contained in 1st Sequence Is Disjoint from 1st Sequence |
| For Permutations Group, Its Element, and Element of Permutations Domain, Sequence of Power Operations of Element on Domain Element Returns Back from Domain Element |
| For Principal Integral Domain and Finite Subset, Sum of Principal Ideals by Elements of Subset Is Principal Ideal by Any of Greatest Common Divisors of Subset |
| For Principal Integral Domain, Rectangle Matrix over Domain, and Invertible Square Matrix over Domain, Sum of Principal Ideals by Specified-Dimensional Subdeterminants of Product Is Sum of Principal Ideals by Same Dimensional Subdeterminants of Rectangle Matrix |
| For Principal Integral Domain, Rectangle Matrix over Domain, and Square Matrix Over Domain, Sum of Principal Ideals by Specified-Dimensional Subdeterminants of Product Is Contained in Sum of Principal Ideals by Same-Dimensional Subdeterminants of Rectangle Matrix |
| 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 Real or Complex Vectors Space with Inner Product, Linear Combination of Finite Vectors Cannot Be Perpendicular to Each Constituent Without Being 0 |
| For Rectangle Matrix over Principal Integral Domain, There Are Some Types of Rows or Columns Operations Each of Which Can Be Expressed as Multiplication by Invertible Matrix from Left or Right |
| For Regular Topological Space, Collapsed Topological Space by Closed Subset Is Hausdorff |
| For Ring and Finite Number of Ideals, Sum of Ideals Is Ideal |
| For Ring, Multiple of 0 Is 0 |
| 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 of Finite Elements, Set of Permutations Has Same Number of Even Permutations and Odd Permutations |
| 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 and 2 Topologies, iff There Is Common Open Cover and Each Open Subset of Each Element of Cover in One Topology Is Open in the Other and Vice Versa, Topologies Are Same |
| For Set of Sequences for Fixed Domain and Codomain, Permutation Bijectively Maps Set onto Set |
| For Set, Intersection of \(\sigma\)-Algebras Is \(\sigma\)-Algebra |
| For Set, Union of Power Set of Set Is Set |
| For Simplicial Complex, Intersection of 2 Affine Simplexes Determined by Subsequences of Ascending Sequences of Barycenters of Faces of Elements of Complex Is Affine Simplex Determined by Intersection of Subsequences |
| For Simplicial Complex, Intersection of 2 Simplexes Is Simplex Determined by Intersection of Sets of Vertexes of Simplexes |
| For Simplicial Complex, Point on Underlying Space Is on Simplex Interior of Unique Simplex |
| For Simplicial Complex, Simplex Interior of Maximal Simplex Does Not Intersect Any Other Simplex |
| For Simplicial Complex, Vertex of Simplex That Is on Another Simplex Is Vertex of Latter Simplex |
| For Simplicial Complex on Finite-Dimensional Real Vectors Space, Each Simplex in Complex Is Faces of Elements of Subset of Maximal Simplexes Set |
| For Simplicial Complex on Finite-Dimensional Real Vectors Space, Open Subset of Underlying Space That Intersects Star Intersects Simplex Interior of Maximal Simplex Involved in Star |
| For Set and 2 Topology-Atlas Pairs, iff There Is Common Chart Domains Open Cover and Each Transition Is Diffeomorphism, Pairs Are Same |
| For Set and Set, Power Set of [Former Set Minus Latter Set] Is [Power Set of Former Set] Elements Minus Latter Set |
| For Set of Sets, Dichotomically Nondisjoint Does Not Necessarily Mean Pair-Wise Nondisjoint |
| For Set, To-Be-Atlas Determines Topology and Atlas |
| For Subset of Topological Space, Closure of Subset Minus Subset Has Empty Interior |
| For Surjection, Preimages of Subsets Are Same iff Subsets Are Same |
| For Topological Space Contained in Ambient Topological Space, if Space Is Ambient-Space-Wise Locally Topological Subspace of Ambient Space, Space Is Topological Subspace of Ambient Space |
| For Topological Space and Finite Number of Open Covers, Intersection of Covers Is Open Cover |
| For Topological Space and Its 2 Products with Euclidean Topological Spaces, Injective Continuous Map Between Products Fiber-Preserving and Linear on Fiber Is Continuous Embedding |
| For Topological Space and Its 2 Products with Euclidean Topological Spaces, Map Between Products Fiber-Preserving and Linear on Fiber Is Continuous iff Canonical Matrix Is Continuous |
| For Topological Space and Locally Finite Set of Closed Subsets, Union of Set Is Closed |
| For Topological Space and Open Cover, Subset Is Open iff Intersection of Subset and Each Element of Open Cover Is Open |
| 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, Point, and Neighborhood of Point, Neighborhood of Point on Neighborhood Is Neighborhood of Point on Base Space |
| For Topological Space, Sequence of Preimages of Natural-Numbers-Closed-Upper-Bounds Intervals Under Exhaustion Function Is Exhaustion of Space by Compact Subsets |
| For Topological Space, Subset of Compact Subset Is Not Necessarily Compact |
| For Topological Space, Subspace, and Subset of Superspace, Subspace Minus Subset as Subspace of Subspace Is Subspace of Superspace Minus Subset |
| For Topological Space, Subspace Subset That Is Compact on Base Space Is Compact on Subspace |
| For Topological Space, Union of Closures of Subsets Is Contained in Closure of Union of Subsets |
| 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 Unique Factorization Domain and Finite Subset, if Greatest Common Divisors of Each Pair Subset of Subset Are Unit Associates, Greatest Common Divisors of Subset Are Unit Associates, but Not Vice Versa |
| For Unique Factorization Domain and Finite Subset, iff Greatest Common Divisors of Each Pair Subset of Subset Are Unit Associates, Least Common Multiples of Subset Are Associates of Multiple of Elements of Subset |
| For Unique Factorization Domain, Method of Getting Greatest Common Divisors of Finite Subset by Factorizing Each Element of Subset with Representatives Set of Associates Quotient Set |
| For Unique Factorization Domain, Method of Getting Least Common Multiples of Finite Subset by Factorizing Each Element of Subset with Representatives Set of Associates Quotient Set |
| For Unique Factorization Domain, if Multiple of Elements Is Divisible by Irreducible Element, at Least 1 Constituent Is Divisible by Irreducible Element |
| For Vectors Bundle and Trivializing Open Subsets Cover, Preimages Under Trivializations of Products of Basis of Open Subset and Basis of \(R^k\) Constitute Basis of Total Space |
| For \(C^\infty\) Vectors Bundle, Chart Open Subset on Base Space Is Not Necessarily Trivializing Open Subset (Probably) |
| For \(C^\infty\) Vectors Bundle, \(C^\infty\) Frame Exists Over and Only Over Trivializing Open Subset |
| For \(C^\infty\) Vectors Bundle, Section over Trivializing Open Subset Is \(C^\infty\) iff Coefficients w.r.t. \(C^\infty\) Frame over There Are \(C^\infty\) |
| For \(C^\infty\) Vectors Bundle, There Is Chart Trivializing Open Cover |
| For \(C^\infty\) Vectors Bundle, Trivialization of Chart Trivializing Open Subset Induces Canonical Chart Map |
| For \(C^\infty\) Vectors Bundle, Trivializing Open Subset Is Not Necessarily Chart Open Subset, but There Is Possibly Smaller Chart Trivializing Open Subset at Each Point on Trivializing Open Subset |
| For Vectors Space and 2 Same-Finite-Dimensional Vectors Subspaces, There Is Common Complementary Subspace |
| For Vectors Space and Linearly Independent Subset, Subset Can Be Expanded to Be Basis |
| For Vectors Space with Inner Product, Set of Nonzero Orthogonal Elements Is Linearly Independent |
| For Vectors Space, Finite Generator Can Be Reduced to Be Basis |
| For Vectors Space, Generator of Space, and Linearly Independent Subset Contained in Generator, Generator Can Be Reduced to Be Basis with Linearly Independent Subset Retained |
| For Vectors Space, Intersection of Finite-Dimensional Subspaces Is Subspace with Dimension Equal to or Smaller than Minimum Dimension of Subspaces |
| For Vectors Space, Subspace, and Complementary Subspace, Finite-Dimensional Subspace That Intersects Complementary Subspace Trivially Is Projected into Subspace as Same-Dimensional Subspace |
| For 'Vectors Spaces - Linear Morphisms' Isomorphism, Image of Linearly Independent Subset or Basis of Domain Is Linearly Independent or Basis on Codomain |
| For Well-Ordered Structure and Its Sub Structure, Ordinal Number of Sub Structure Is Member of or Is Ordinal Number of Base Structure |
| For (n + n') x (n + n'') Injective Matrix with Right-Top n x n'' Submatrix 0, Matrix with Left-Top n x n Submatrix Replaced with Injective Matrix Is Injective |
| For n x n Matrix, if There Are m Rows with More Than n - m Same Columns 0, Matrix Is Not Invertible |
| For n-Symmetric Group and n-Cycle, Centralizer of Cycle on Symmetric Group Is Cyclic Group by Cycle |
| Formalization of Local Slice Condition for Embedded Submanifold or Local Slice Condition for Embedded Submanifold with Boundary |
| 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 |
| Functor Maps Isomorphism to Isomorphism |
| 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 |
| Group Is 'Groups - Homomorphisms' Isomorphic to Reversed Operator Group of Group |
| Group as Direct Sum of Finite Number of Normal Subgroups Is Group as Direct Sum of Any Reordered and Combined Normal Subgroups |
| Group as Direct Sum of Finite Number of Normal Subgroups Is 'Groups - Homomorphisms' Isomorphic to Direct Product of Subgroups |
| 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 from Subset of Euclidean \(C^\infty\) Manifold or Closed Upper Half Euclidean \(C^\infty\) Manifold with Boundary into Subset of Euclidean \(C^\infty\) Manifold or Closed Upper Half Euclidean \(C^\infty\) Manifold with Boundary Is \(C^\infty\) |
| 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 |
| In Order to Check Continuousness of Map, Preimages of Only Basis or Subbasis Are Enough |
| 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 Group Homomorphism Is 'Groups - Homomorphisms' Isomorphism onto Range |
| Injective Map Between \(C^\infty\) Manifolds with Boundary Is \(C^\infty\) Embedding, if Domain Restriction of Map on Each Element of Open Cover Is \(C^\infty\) Embedding onto Open Subset of Range or Codomain |
| 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 |
| Integers Ring Is Principal Integral Domain |
| 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 Minus Set and Set Is Intersection of 1st Set and 3rd Set Minus Intersection of 2nd Set and 3rd Set |
| Intersection of Set of Transitive Relations Is Transitive |
| Intersection of Simplicial Complexes Is Simplicial Complex, and Underlying Space of Intersection Is Contained in but Not Necessarily Equal to Intersection of Underlying Spaces of Constituents |
| Intersection of Subgroup of Group and Normal Subgroup of Group Is Normal Subgroup of Subgroup |
| Intersection of Subsets Is Complement of Union of Complements of Subsets |
| Intersection of Union of Subsets and Subset Is Union of Intersections of Each of Subsets and Latter Subset |
| 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 |
| Latin Square with Each Row Regarded as Permutation Forms Group iff Composition of 2 Rows Is Row, and Group's Multiplications Table Is Generated by Certain Way from Square |
| 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 Injection Between Same-Finite-Dimensional Vectors Spaces Is 'Vectors Spaces - Linear Morphisms' Isomorphism |
| Linear Range 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 \(C^\infty\) Manifolds with Boundary Is \(C^k\) if and Only if Domain Restriction of Map to Each Element of Open Cover Is \(C^k\) |
| Map Between Groups That Maps Product of 2 Elements to Product of Images of Elements Is Group Homomorphism |
| 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 |
| 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 from Open Subset of \(C^\infty\) Manifold with Boundary onto Open Subset of Euclidean \(C^\infty\) Manifold or Closed Upper Half Euclidean \(C^\infty\) Manifold with Boundary Is Chart Map iff It Is Diffeomorphism |
| Map from Topological Space into Finite Product Topological Space Is Continuous iff All Component Maps 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 Is Bijection iff Preimage of Codomain Point Is 1 Point Subset |
| 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 Subset Minus Subset Is Preimage of 1st Subset Minus Preimage of 2nd Subset |
| 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 |
| Matrices Multiplications Map Is Continuous |
| Maximal Element of Set w.r.t. Inverse of Ordering Is Minimal Element of Set w.r.t. Original Ordering |
| Memorandum on Powers of Group, Ring, or Field Elements |
| 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 |
| Motion Is Injective |
| 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 |
| Normal Subgroup of Group Is Normal Subgroup of Subgroup of Group Multiplied by Normal Subgroup |
| Norms on Finite-Dimensional Real Vectors Space Are Equivalent |
| On 2nd-Countable Topological Space, Open Cover Has Countable Subcover |
| On Set of Continuous Maps Between Topological Spaces, Being Homotopic Is Equivalence Relation |
| 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 Subset of \(C^\infty\) Trivializing Open Subset Is \(C^\infty\) Trivializing Open Subset |
| 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 |
| Orthogonal Linear Map Between Same-Finite-Dimensional Normed Vectors Spaces Is 'Vectors Spaces - Linear Morphisms' Isomorphism and Inverse Is Orthogonal Linear Map |
| Orthogonal Linear Map Is Motion |
| Over Field, Polynomial and Nonzero Polynomial Divisor Have Unique Quotient and Remainder |
| Over Field, n-Degree Polynomial Has at Most n Roots |
| 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 |
| Permutation Bijectively Maps Set of Permutations onto Set of Permutations by Composition from Left or Right |
| 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 |
| Polynomials Ring over Integral Domain Is Integral Domain |
| 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 |
| Principal Integral Domain Is Greatest Common Divisors Domain, and for 2 Elements, Each of Greatest Common Divisors Is One by Which Sum of Principal Ideals by 2 Elements Is Principal Ideal |
| 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 Hausdorff Topological Spaces Is Hausdorff |
| 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 |
| Projection from Vectors Space into Subspace w.r.t. Complementary Subspace Is Linear Map and Image of Any Subspace Is Subspace |
| Projective Hyperplane Is Hausdorff |
| Proposition 1 or Proposition 2 iff if Not Proposition 2, Proposition 1 |
| Pushforward Image of \(C^\infty\) Vectors Field Along Curve on Regular Submanifold into Supermanifold Under Inclusion Is \(C^\infty\) |
| Quotient Ring of Commutative Ring by Ideal Is Commutative Ring |
| Quotient Ring of Integers Ring by Prime Principal Ideal Is Field |
| 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 |
| Range Under Lie Algebra Homomorphism Is Lie Sub-Algebra of Codomain |
| Range of Group Homomorphism Is Subgroup of Codomain |
| 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 ... |
| Restricted \(C^\infty\) Vectors Bundle W.r.t. Embedded Submanifold with Boundary Is Embedded Submanifold with Boundary |
| 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 |
| Retract of Hausdorff Topological Space Is Closed |
| 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 Minus (Set Minus Set) Is Not Necessarily but Contains (1st Set Minus 2nd Set) Minus 3rd Set |
| Set Minus (Set Minus Set) Is Union of 1st Set Minus 2nd Set and Intersection of 1st Set and 3rd Set |
| Set of \(C^\infty\) Sections of \(C^\infty\) Vectors Bundle Linearly Independent at Point Is Linearly Independent on Open Neighborhood of Point |
| 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 Interior of Affine Simplex Is Open on Affine Simplex with Canonical Topology |
| Smith Normal Form Theorem for Rectangle Matrix over Principal Integral Domain |
| Standard 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 |
| Subgroup of Group Multiplied by Normal Subgroup of Group Is Subgroup of Group |
| 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 Affine-Independent Set of Points on Real Vectors Space Is Affine-Independent |
| Subset of 1st Category Subset Is of 1st Category |
| Subset of Not-Necessarily-Open Topological Subspace Is Open on Subspace If It Is Open on Basespace |
| 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 of Underlying Space of Finite Simplicial Complex on Finite-Dimensional Real Vectors Space Is Closed iff Its Intersection with Each Element of Complex Is Closed |
| 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 |
| Tangent Vector at Point on \(C^\infty\) Manifold with Boundary Is Velocity of \(C^\infty\) Curve, Especially from Half Closed Interval, Especially as Linear in Coordinates |
| 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 |
| 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 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 |
| Union of Set Minus Set and Set Is Not Necessarily but Contains Union of 1st Set and 3rd Set Minus Union of 2nd Set and 3rd Set |
| Union of Subsets Is Complement of Intersection of Complements of Subsets |
| 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 Convex Set Spanned by Non-Affine-Independent Set of Base Points on Real Vectors Space Is Affine Simplex, It Is Spanned by Affine-Independent Subset of Base Points |
| When Convex Set Spanned by Non-Affine-Independent Set of Base Points on Real Vectors Space Is Affine Simplex, Point Whose Original Coefficients Are All Positive Is on Simplex Interior of Simplex, but Point One of Whose Original Coefficients Is 0 Is Not Necessarily on Simplex Boundary of Simplex |
| 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 |
