ExampleConstruction: AI-adaptedVerification: AI-generatedprecheck passaudited 2026-07-31

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Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
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A one-point space and $R$ are homotopy equivalent but not homeomorphic

Example

Let $P = {p}$ be a one-point space. The maps $i : P \to R$ , $i (p) = 0$ , and $q : R \to P$ exhibit $P ≃ R$ , although the spaces are not homeomorphic.

Facts & Assumptions

Given: The one-point space $P$ and the real line.

[L1]

The real line contracts to $0$ by $H (x, t) = (1 - t) x$ (Every nonempty interval and every $R^{n}$ with $n \geq 1$ contracts to any chosen point).

[L2]

A homotopy equivalence has a homotopy inverse whose composites are homotopic to the identity maps (Homotopy equivalences, homotopy inverses and spaces of the same homotopy type).

[L3]

A homeomorphism is a bijection, and $R$ is uncountable (Homeomorphism, open map, closed map, embedding, and what it means for a property to be topological, $R$ is uncountable (Cantor's nested intervals, 1874)).

[L4]

The refutation in FALSE: homotopy-equivalent spaces must be homeomorphic uses this same pair as its counterexample.

Verification

technique · direct

1.1

Both $i$ and $q$ are continuous, since the preimage of any open set is either empty or the entire one-point source or target preimage.

construct

1.2

No bijection exists from the finite set $P$ to the uncountable set $R$ , so no homeomorphism exists by [L3].

2.1

One has $q \circ i = {id}_{P}$ and $i \circ q = c_{0} ≃ {id}_{R}$ by [L1]. Hence $i$ and $q$ are homotopy inverses by [L2].

step 1.1L1L2

3.1

Steps 2.1 and 1.2 verify the claimed contrast, agreeing with [L4].

step 2.1step 1.2L4∎

Depends on

Used by

Nothing in the library uses this result yet.

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)

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