Example 4

Let E be the numerical space of $n$ dimensions (or, more generally, a topological vector space over $\mathbf{R}$) and let X be a subset of E. One says that the set X is star-shaped at a point $x$ of X if, for all $y\in X$ and all $t\in \mathbf{I}$, the point $tx+ (1-t)y$ belongs to X. A convex subset (I, p. 122) of E is star-shaped at each of its points.

A topological subspace X of E which is star-shaped at one of its points $x$ is contractible at this point. Indeed, the mapping $\sigma : X\times \mathbf{I}\rightarrow X$ defined by $\sigma (y, t) =tx+ (1-t)y$ is a pointed homotopy at $x$ joining Id$_X$ to the constant mapping with image $\{x\}$.

In particular, every interval of $\mathbf{R}$, every convex subset of a numerical space or, more generally, of a topological vector space over $\mathbf{R}$ is contractible at each of its points.