ExampleConstruction: AI-adaptedVerification: AI-generatedverified 2026-09-26 (gpt-6-sol)

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

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 map $H : R \times [0, 1] \to R$ , $H (x, t) = (1 - t) x$ , is continuous, with $H (x, 0) = x$ and $H (x, 1) = 0$ .

[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

1.1construct

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.

1.2L3

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

2.1step 1.1L1L2

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].

3.1step 2.1step 1.2L4∎

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

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

29 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