CorollaryStatement: Literature-sourcedProof: AI-adaptedprecheck passaudited 2026-07-31verified 2026-08-08 (gpt-5.6-terra-codex-subscription)

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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.

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Every nonempty convex subset of $R^{n}$ is contractible

Statement

Let $n \geq 1$ . Every nonempty convex subset $C \subseteq R^{n}$ , with its Euclidean subspace topology, is contractible. More precisely, for each $c \in C$ the formula

$H (x, t) = (1 - t) x + t c$

is a homotopy from ${id}_{C}$ to the constant map at $c$ .

Facts & Assumptions

Given: A nonempty convex subset $C \subseteq R^{n}$ and a point $c \in C$ .

[L1]

Any two continuous maps into a nonempty convex subset of $R^{n}$ are homotopic by the straight-line formula (Any two continuous maps into a nonempty convex subset of $R^{n}$ are homotopic by straight lines).

[L2]

A nonempty space is contractible exactly when its identity map is nullhomotopic (A nonempty space is contractible if and only if its identity map is nullhomotopic).

Proof

technique · direct

1.1

Apply [L1] to ${id}_{C} : C \to C$ and the constant map $c_{c} : C \to C$ . The resulting homotopy is $H (x, t) = (1 - t) x + t c$ .

2.1

Thus ${id}_{C}$ is nullhomotopic, so $C$ is contractible by [L2].

step 1.1L2∎

Depends on

Used by

Dependency tree · two levels

6 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

Algebraic Topology lecture notes (UC Riverside) (standard reference, not scraped)

Every nonempty convex subset of RnRn is contractible