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

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Any two continuous maps into a nonempty convex subset of $R^{n}$ are homotopic by straight lines

Statement

Let $n \geq 1$ , let $C \subseteq R^{n}$ be nonempty and convex in the sense stated in For continuous maps into a convex subset of $R^{n}$ , the straight-line formula defines a continuous homotopy, and let $f, g : X \to C$ be continuous. Then

$H (x, t) = (1 - t) f (x) + t g (x)$

is a homotopy from $f$ to $g$ .

Facts & Assumptions

Given: A nonempty convex $C \subseteq R^{n}$ with $n \geq 1$ and continuous maps $f, g : X \to C$ .

[L1]

The straight-line formula defines a continuous map $H : X \times I \to C$ (For continuous maps into a convex subset of $R^{n}$ , the straight-line formula defines a continuous homotopy).

[A1]

A homotopy from $f$ to $g$ is a continuous $H : X \times I \to C$ with $H (x, 0) = f (x)$ and $H (x, 1) = g (x)$ (Homotopies of continuous maps, homotopies relative to a subspace, and path homotopies relative to the endpoints).

Proof

technique · direct

1.1

The map $H (x, t) = (1 - t) f (x) + t g (x)$ is continuous by [L1].

1.2

Substitution gives $H (x, 0) = f (x)$ and $H (x, 1) = g (x)$ for every $x \in X$ .

algebra

2.1

Thus $H$ satisfies the continuity and endpoint conditions of [A1], so it is a homotopy from $f$ to $g$ .

step 1.1step 1.2A1∎

Depends on

Used by

Dependency tree · two levels

11 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)

Any two continuous maps into a nonempty convex subset of RnRn are homotopic by straight lines