CorollaryStatement: AI-adaptedProof: AI-generatedSession-authored (Fable 5 assisted)precheck passaudited 2026-07-31

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Homotopy relative to a fixed subspace, and path homotopy relative to endpoints, are equivalence relations

Statement

For fixed spaces $X,Y$ and a fixed subspace $A\subseteq X$ , the relation $\simeq_A$ is an equivalence relation on the set of continuous maps $X\to Y$ that have a prescribed restriction to $A$ . In particular ordinary homotopy is an equivalence relation on the continuous maps $X\to Y$ .

For fixed endpoints $y_0,y_1\in Y$ , path homotopy relative to the endpoints is an equivalence relation on the set of paths from $y_0$ to $y_1$ .

Facts & Assumptions

Given: Spaces $X,Y$ , a subspace $A\subseteq X$ , and the relations defined in Homotopies of continuous maps, homotopies relative to a subspace, and path homotopies relative to the endpoints.

[L1]

Homotopy rel $A$ is reflexive and symmetric (Homotopy relative to a subspace is reflexive and symmetric).

[L2]

Homotopy rel $A$ is transitive by the two-piece reparametrisation construction (Two homotopies relative to the same subspace concatenate after piecewise-linear reparametrisation).

[L3]

A relation is an equivalence relation exactly when it is reflexive, symmetric and transitive (Equivalence relation, equivalence class, and the quotient set $A/{\sim}$ ).

Proof

technique · direct

1.1

By [L1] and [L2], $\simeq_A$ is reflexive, symmetric and transitive, so it is an equivalence relation by [L3]. Taking $A=\varnothing$ gives ordinary homotopy.

L1L2L3

2.1

Paths from $y_0$ to $y_1$ are continuous maps $I\to Y$ with one prescribed restriction to the subspace $\{0,1\}$ , and their path homotopies are exactly homotopies rel $\{0,1\}$ . Hence step 1.1 applies to them.

step 1.1given∎

Depends on

Used by

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 60 results over 18 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources

A. Hatcher, Algebraic Topology, Section 0 (standard reference, not scraped)
Homotopy lecture notes (University of Padua) (standard reference, not scraped)