ExampleConstruction: AI-adaptedVerification: AI-generatedSession-authored (Fable 5 assisted)precheck passaudited 2026-07-31verified 2026-08-08 (gpt-5.6-terra-codex-subscription)

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The formula $H(x,t)=(1-t)f(x)+tg(x)$ gives an explicit homotopy between maps into $\mathbb{R}^n$

Example

Let $n\ge1$ , let $X$ be a topological space, and let $f,g:X\to\mathbb R^n$ be continuous. Since $\mathbb R^n$ is convex, the straight-line formula

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

deforms $f$ to $g$ .

Facts & Assumptions

Given: Continuous maps $f,g:X\to\mathbb R^n$ with $n\ge1$ .

[L1]

The straight-line formula is continuous for maps into a convex subspace of $\mathbb R^n$ (For continuous maps into a convex subset of $\mathbb{R}^n$ , the straight-line formula defines a continuous homotopy).

[L2]

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

Verification

technique · direct

1.1

For $u,v\in\mathbb R^n$ and $t\in I$ , the vector $(1-t)u+tv$ lies in $\mathbb R^n$ , so $\mathbb R^n$ is convex.

algebra

1.2

The map $H$ is continuous by [L1].

2.1

Substitution gives $H(x,0)=f(x)$ and $H(x,1)=g(x)$ , so [L2] identifies $H$ as a homotopy from $f$ to $g$ .

step 1.1step 1.2L2∎

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: 56 results over 15 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

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

The formula H(x,t)=(1−t)f(x)+tg(x)H(x,t)=(1-t)f(x)+tg(x)H(x,t)=(1−t)f(x)+tg(x) gives an explicit homotopy between maps into Rn\mathbb{R}^nRn