False statementConstruction: AI-adaptedVerification: AI-generatedSession-authored (Fable 5 assisted)precheck 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 $\mathbb R$ 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 $g\circ f\simeq\operatorname{id}_X$ and $f\circ g\simeq\operatorname{id}_Y$ (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 $\mathbb{R}^n$ 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 $\mathbb R$ is uncountable, whereas a singleton is finite ( $\mathbb{R}$ is uncountable (Cantor's nested intervals, 1874)).

Refutation

technique · direct

1.1

Define $i:P\to\mathbb R$ by $i(p)=0$ and let $q:\mathbb R\to P$ 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 $P\to\mathbb R$ exists by [L3], so no homeomorphism exists by [L2].

L2L3

2.1

One has $q\circ i=\operatorname{id}_P$ , while $i\circ q=c_0\simeq\operatorname{id}_{\mathbb R}$ by [L1]. Thus $i$ is a homotopy equivalence with homotopy inverse $q$ by [A1].

step 1.1L1A1

3.1

Therefore $P$ and $\mathbb R$ 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 · next 3 levels

Direct dependencies and their dependencies through the next three levels: 69 results over 22 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)