Alphabeta MathHomotopy and Homotopy Equivalence
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The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.
- Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
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✓ 18 results · all verified · 0 also independently AI-judged
Every result on this page is machine-checked by a proof checker and read in full and owner-audited; the judge is an additional, independent cross-model AI review of the proofs. The 18 not AI-judged were verified by owner audit (typically over a confirmed judge false positive), not failures.
1 · Prerequisites
Abstract Algebra
Linear Algebra
Real Analysis
- Construction of the Real Numbers via Cauchy Sequences
- Construction of the Real Numbers via Dedekind Cuts
- Continuity, IVT, EVT, and Uniform Continuity
- Countability and Uncountability
- Foundations of the Real Numbers for Analysis
- Limits of Real Functions
- Monotone Sequences, Bolzano-Weierstrass, and Cauchy Completeness
- Rⁿ as a Normed Space; Vector-Valued Functions
- Roots, Rational Powers, and Classical Inequalities
- Sequences and Limits
- Suprema and Infima
- Topology of ℝ
Set Theory
2 · Logical flowchart
3 · Items
Definition
Homotopies of continuous maps, homotopies relative to a subspace, and path homotopies relative to the endpoints
Lemma
Homotopy relative to a subspace is reflexive and symmetric
Lemma
Two homotopies relative to the same subspace concatenate after piecewise-linear reparametrisation
Corollary
Homotopy relative to a fixed subspace, and path homotopy relative to endpoints, are equivalence relations
Theorem
Precomposition and postcomposition by continuous maps preserve homotopies, including their relative form
Definition
Nullhomotopic maps and contractible spaces
Corollary
A nonempty space is contractible if and only if its identity map is nullhomotopic
Definition
Homotopy equivalences, homotopy inverses and spaces of the same homotopy type
Theorem
Having the same homotopy type is an equivalence relation on topological spaces
Corollary
A continuous map homotopic to a homotopy equivalence is itself a homotopy equivalence
Definition
Retractions and deformation retracts, with a deformation retraction required to fix the retract pointwise
Theorem
The inclusion of a deformation retract is a homotopy equivalence with the retraction as homotopy inverse
Corollary
Every nonempty retract of a contractible space is contractible
Corollary
A homotopy equivalence induces a bijection between path components
Lemma
For continuous maps into a convex subset of ℝⁿ, the straight-line formula defines a continuous homotopy
Theorem
Any two continuous maps into a nonempty convex subset of ℝⁿ are homotopic by straight lines
Corollary
Every nonempty convex subset of ℝⁿ is contractible
Corollary
Every nonempty contractible space is path-connected
Theorem
For n≥1, radial normalisation is a deformation retraction of ℝⁿ∖{0} onto Sⁿ⁻¹
Corollary
For n≥1, the punctured Euclidean space ℝⁿ∖{0} is homotopy equivalent to Sⁿ⁻¹
False statement
FALSE: every retract is a deformation retract
False statement