Exercise 2, § 1

Let B be a subset of $\mathbf{R}^n$, let $a$ be an interior point of B; denote by S the boundary of B. Suppose that B is star-shaped with respect to $a$ and that every half-line with origin $a$ meets S at a unique point. Let $f:\mathbf{R}_+^*\times S\rightarrow \mathbf{R}^n-\{a\}$ be the mapping defined by $f(t, y) =ty+ (1-t)a$. Exercise prove that $f$ is a homeomorphism such that $f(]0,1]\times S) = B-\{a\}$ and $f(\{1\} \times S) = S$.

This result applies in particular when B is a convex, closed and bounded set of $\mathbf{R}^n$ and $a$ is an adherent point of B (cf. I, p. 123, prop. 2 and EVT, II, p. 15, prop. 16).