False statementConstruction: AI-adaptedVerification: AI-generatedprecheck passaudited 2026-07-31

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FALSE: homotopy-equivalent spaces must be homeomorphic

Statement

False claim. If two topological spaces are homotopy equivalent, then they are homeomorphic.

Facts & Assumptions

Given: A one-point space $P = {p}$ and the real line $R$ with its usual topology.

[A1]

A map $f : X \to Y$ is a homotopy equivalence when it has a continuous $g : Y \to X$ with $g \circ f ≃ {id}_{X}$ and $f \circ g ≃ {id}_{Y}$ (Homotopy equivalences, homotopy inverses and spaces of the same homotopy type).

[L1]

The real line is a nonempty convex subset of itself, hence is contractible, and its identity is homotopic to the constant map at $0$ (Every nonempty convex subset of $R^{n}$ is contractible, A nonempty space is contractible if and only if its identity map is nullhomotopic).

[L2]

A homeomorphism is in particular a bijection (Homeomorphism, open map, closed map, embedding, and what it means for a property to be topological).

[L3]

The set $R$ is uncountable, whereas a singleton is finite ( $R$ is uncountable (Cantor's nested intervals, 1874)).

Refutation

technique · direct

1.1

Define $i : P \to R$ by $i (p) = 0$ and let $q : R \to P$ be the unique map. Both are continuous: the preimage of an open set under either map is either empty or the whole domain.

construct

1.2

No bijection $P \to R$ exists by [L3], so no homeomorphism exists by [L2].

L2L3

2.1

One has $q \circ i = {id}_{P}$ , while $i \circ q = c_{0} ≃ {id}_{R}$ by [L1]. Thus $i$ is a homotopy equivalence with homotopy inverse $q$ by [A1].

step 1.1L1A1

3.1

Therefore $P$ and $R$ are homotopy equivalent by step 2.1 but not homeomorphic by step 1.2, refuting the claim.

step 2.1step 1.2∎

Depends on

Used by

A one-point space and ℝ are homotopy equivalent but not homeomorphic Example

Dependency tree · two levels

31 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

A. Hatcher, Algebraic Topology, Section 0 (standard reference, not scraped)