Alphabeta MathArticle items · 01 — A nonempty space is contractible if and only if its identity map is nullhomotopic
Presents 26 items from the article and its immediate prerequisites, covering published library prerequisites.
1 · Prerequisites
- Nullhomotopic maps and contractible spaces
- Precomposition and postcomposition by continuous maps preserve homotopies, including their relative form
- Paths, path-connected spaces and path components
- A map into a product is continuous iff each of its components is; the projections are continuous and open; and each projection is surjective when every factor is nonempty, which for an infinite index set uses the Axiom of Choice
- For a map of spaces the following agree: continuity at every point, preimages of open sets open, preimages of closed sets closed, preimages of subbasic open sets open, and f(Ā) ⊆ f(A)
- Linear map between vector spaces over the same field
- The space L(V,W) of linear maps with pointwise addition and scalar multiplication
- Free group on a set of generators
- Group presentation by generators and relations
- Relators and relations; finitely generated, finitely related, and finite presentations
- The normal closure of a subset of a group
- The quotient group G/N and coset product (gN)(hN)=ghN
- The natural numbers ℕ (von Neumann)
- The well-ordering principle
- Group and abelian group
- Coxeter matrices, the presented Coxeter group, reduced words, length, and standard parabolic subgroups
- The Bruhat graph by length-increasing reflection chains, the Bruhat order, inversion symmetry, and reflection parity
- Choice function
- Homotopies of continuous maps, homotopies relative to a subspace, and path homotopies relative to the endpoints
- Homotopy relative to a fixed subspace, and path homotopy relative to endpoints, are equivalence relations
- Words in an alphabet with formal inverses, elementary cancellation, and reduced words
- Semigroup and monoid
- Equivalence relation, equivalence class, and the quotient set A/∼
- The subgroup ⟨ S ⟩ generated by a subset, the cyclic subgroup ⟨ g ⟩, and cyclic groups
- The real Coxeter form, its radical, reflections, and form-preserving maps
- Reflections: involutivity, form invariance, fixed hyperplane, and exact rank-two order
- Monoid homomorphism and group homomorphism
- Invertible linear maps, linear isomorphisms, and inverse linear maps
- Identity maps and composites of linear maps are linear
- The invertible elements of a monoid form a group under the restricted operation
- Linear combination of a finite list, and the span span(S) as the smallest linear subspace containing S
- Adjacency, incidence, open and closed neighbourhoods, vertex degree, minimum degree and maximum degree
- Walks, closed walks, trails, paths and cycles, with length equal to the number of traversed edges
- Positive and negative definiteness, the inertia (p,q,r), rank p+q, and signature p-q of a real symmetric bilinear or quadratic form
- The matrix, left and right radicals, rank, and nondegeneracy of a bilinear form on a finite-dimensional space
- The quotient vector space V/W and its canonical projection
- Real and complex inner-product spaces and their induced length
- Affine subspaces as translates x+U of linear subspaces
- Linear subspace of a vector space
- Basis of a vector space: a linearly independent spanning subset; and ordered basis: an injective finite list whose image is a basis
- Bilinear forms on V correspond linearly and bijectively to linear maps V→ V^*
- Support, intrinsic parabolic presentations, minimal coset representatives and length additivity, with the type-A identification
- Length parity, exchange, two-letter deletion, and faithfulness of the signed reflection action
- Graph isomorphisms, automorphisms and graph complements
- Classification of finite Coxeter systems, including the H and dihedral families
- Vector space over a field
- The complex numbers as ℝ[x]/(x²+1), with the real embedding and imaginary unit i
- Universal property of the tensor product for balanced maps into abelian groups
- A union of connected subspaces with a point in common is connected, and so is a union of a family in which every member meets a fixed connected member
- Topology on a set, open and closed sets, clopen sets, the closed-set axiomatisation, and the coarser/finer comparison
- Subspace topology: the traces of the open sets, its closed sets and its bases, the continuity of the inclusion, and the characteristic property of a map into a subspace
- The discrete, indiscrete, cofinite, cocountable, particular-point and Sierpinski topologies
- Unless stated otherwise, graph means finite, simple and undirected; orders, sizes and empty-set conventions are fixed here
- Subgraphs, induced subgraphs and spanning subgraphs
- Every walk between two vertices contains a path between the same endpoints
- Interior, closure, boundary, exterior, derived set and isolated point in a topological space
- A point lies in the closure of A iff every basic neighbourhood of it meets A; the closure is the smallest closed superset and equals A together with its derived set
- Separated sets, disconnection, and connected subset of ℝ
- Neighbourhood of a point and neighbourhood base, with this library's convention that a neighbourhood need not be open
- Injection, surjection, bijection
- A finite list v : n → V is an ordered basis if and only if every x ∈ V equals ∑_i<n λᵢ vᵢ for exactly one λ : n → F; those scalars are the coordinates of x in that ordered basis
- The vector space M_m × n(F) := F^ m × n of m by n matrices over a field, with entrywise operations
- Subgroup
- Homeomorphism, open map, closed map, embedding, and what it means for a property to be topological
- The disjoint union (coproduct) ⨆ᵢ Xᵢ with the final topology of the canonical injections: a set is open exactly when each of its traces is
- Left group actions, transitive actions, and faithful actions
- A free group action has no nonidentity element fixing a point
- Maps and isomorphisms of covering spaces over a fixed base
- The circle as S¹=ℝ/ℤ with basepoint [0]
- p:ℝ→ℝ/ℤ is a covering map with translated interval sheets
- Lifts of maps, paths, and homotopies through a covering map
- Existence and uniqueness of path lifts through a covering map
- If V has a basis with n elements and a basis with m elements then n = m; and if V has one finite basis then every basis of V is finite
- Linear independence: a finite list v : n → V is independent when ∑_i<n λᵢ vᵢ = 0_V forces every λᵢ = 0_F, and a subset S ⊆ V is independent when every injective finite list into S is independent
- A field is a vector space over itself, and over any subfield K ⊆ F every F-vector space is a K-vector space by restricting the scalars
- Subfield: a subring of a field closed under inverses of its nonzero elements, and therefore a field with the restricted operations
- Field
- Finite, countably infinite, countable, uncountable
- Equinumerous sets, A ≈ B and A ⪯ B
- The pigeonhole principle on ℕ
2 · Logical flowchart
Arrows show item dependencies. A grouped prerequisite node points to items that use one or more of its members; their exact links are listed above.
3 · Items
CorollaryLibrary prerequisite
A nonempty space is contractible if and only if its identity map is nullhomotopic
CorollaryLibrary prerequisite
Every nonempty contractible space is path-connected
DefinitionLibrary prerequisite
Linear functionals and the algebraic dual V^*=L(V,F)
DefinitionLibrary prerequisite
Arbitrary-rank Coxeter presentations, length, reflections, and Bruhat order
DefinitionLibrary prerequisite
The Axiom of Choice
DefinitionLibrary prerequisite
Based loops and the fundamental group
DefinitionLibrary prerequisite
Cell attachment by a characteristic map
DefinitionLibrary prerequisite
Artin monoid and Artin group presentations, and the canonical monoid-to-group map
DefinitionLibrary prerequisite
The canonical reflection homomorphism, roots, reflections, and the positive cone
DefinitionLibrary prerequisite
Coxeter diagrams: edges, labels, components and finite type
DefinitionLibrary prerequisite
Irreducible affine Coxeter type: the corank-one form, the radical quotient, and the affine slice
DefinitionLibrary prerequisite
Standard parabolic subgroups, descent-free one- and two-sided representatives, parabolic and reflection subgroups
DefinitionLibrary prerequisite
The standard affine diagrams A-tilde, B-tilde, C-tilde, D-tilde, E-tilde, F-tilde and G-tilde
DefinitionLibrary prerequisite
Complexification as ℂ⊗_ℝV with its canonical real-linear embedding
DefinitionLibrary prerequisite
Connected components, quasicomponents, and totally disconnected spaces
DefinitionLibrary prerequisite
Connected graphs and connected components defined by the existence of vertex paths
DefinitionLibrary prerequisite
Separation of a topological space, connected and disconnected spaces, clopen sets, and connected subsets
DefinitionLibrary prerequisite
Continuity of a map of topological spaces at a point and globally
DefinitionLibrary prerequisite
Coordinate columns [v]_B and matrices [T]_B^C of linear maps relative to ordered bases
DefinitionLibrary prerequisite
Left and right cosets gH and Hg of a subgroup
DefinitionLibrary prerequisite
Covering maps, evenly covered neighbourhoods, fibres, sheets, and trivial coverings
DefinitionLibrary prerequisite
Covering-space actions by disjoint translates of neighbourhoods
DefinitionLibrary prerequisite
CW complex with closure finiteness and weak topology
DefinitionLibrary prerequisite
Deck transformations and the deck-transformation group of a covering
DefinitionLibrary prerequisite
The degree of a based circle loop
DefinitionLibrary prerequisite