ExampleConstruction: AI-adaptedVerification: AI-generatedSession-authored (Fable 5 assisted)precheck passaudited 2026-07-31

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A one-point space and $\mathbb{R}$ are homotopy equivalent but not homeomorphic

Example

Let $P=\{p\}$ be a one-point space. The maps $i:P\to\mathbb R$ , $i(p)=0$ , and $q:\mathbb R\to P$ exhibit $P\simeq\mathbb R$ , 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 $\mathbb{R}^n$ with $n\ge1$ 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 $\mathbb R$ is uncountable (Homeomorphism, open map, closed map, embedding, and what it means for a property to be topological, $\mathbb{R}$ 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 $\mathbb R$ , so no homeomorphism exists by [L3].

2.1

One has $q\circ i=\operatorname{id}_P$ and $i\circ q=c_0\simeq\operatorname{id}_{\mathbb R}$ 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 · next 3 levels

Direct dependencies and their dependencies through the next three levels: 70 results over 23 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)

A one-point space and R\mathbb{R}R are homotopy equivalent but not homeomorphic