TheoremStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-03

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Every nonempty convex subset of $\mathbb R^n$ is simply connected

Statement

Let $n\geq1$ and let $C\subseteq\mathbb R^n$ be nonempty and convex, with its Euclidean subspace topology. Then $C$ is simply connected. More explicitly, for every basepoint $x_0\in C$ and every loop $\alpha$ at $x_0$ , the formula

$H(s,t)=(1-t)\alpha(s)+t x_0$

is a path homotopy relative to the endpoints from $\alpha$ to the constant loop at $x_0$ .

Facts & Assumptions

Given: A nonempty convex subset $C\subseteq\mathbb R^n$ , a basepoint $x_0\in C$ , and a based loop $\alpha:I\to C$ .

[L1]

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

[L2]

Every nonempty contractible space is path-connected, and the published straight-line contraction makes a nonempty convex subset contractible (Every nonempty contractible space is path-connected and its dependency Every nonempty convex subset of $\mathbb{R}^n$ is contractible).

[L3]

A loop class is the identity exactly when the loop is endpoint-homotopic to the constant loop (Based loops and the fundamental group, Loop classes form the group $\pi_1(X,x_0)$ under concatenation).

[L4]

Simple connectedness means nonempty path-connectedness and a one-element fundamental group at every basepoint (Simply connected topological spaces).

Proof

technique · direct

1.1

Apply [L1] to the maps $\alpha:I\to C$ and $c_{x_0}:I\to C$ ; it gives the displayed continuous homotopy $H(s,t)=(1-t)\alpha(s)+t x_0$ .

2.1

Since $\alpha(0)=\alpha(1)=x_0$ , one has $H(0,t)=x_0=H(1,t)$ for every $t$ , so this homotopy is relative to the endpoints.

step 1.1L3

3.1

Steps 1.1 and 2.1 show that every loop at every basepoint represents the constant-loop class, so each fundamental group has one element; [L2] supplies nonempty path-connectedness.

step 1.1step 2.1L2L3

4.1

Therefore $C$ is simply connected by [L4].

step 3.1L4∎

Depends on

Used by

The fundamental group of the unit interval is trivial at every basepoint Example

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 85 results over 18 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, Chapter 1, Example 1.4 (standard reference, not scraped)

Every nonempty convex subset of Rn\mathbb R^nRn is simply connected