Example 3

A subset of a numerical space (or, more generally, a topological vector space over $\mathbf{R}$) which is star-shaped at one of its points is path-connected.

Let X be a topological space. We denote by $\pi_0(X)$ the set of path-connected components of X. Let P be a topological space reduced to a point. The mapping from X into [P; X] which, to every point $x$ of X, associates the homotopy class $\varphi_x$ of the mapping $f_x: P\rightarrow X$ having image $x$, defines by passing to the quotient a bijection, called canonical, of $\pi_0(X)$ onto [P; X]. We have $\pi_0(\emptyset ) =\emptyset$. For a nonempty topological space to be path-connected, it is necessary and sufficient that $\pi_0(X)$ be a set with one element.

Let X and Y be topological spaces and $f: X\rightarrow Y$ a continuous mapping. The image $f(C)$ of every path-connected component C of X is path-connected (III, p. 258, prop. 3), hence contained in a path-connected component of Y. We denote by $\pi_0(f):\pi_0(X)\rightarrow \pi_0(Y)$ the mapping which, to a path-connected component C of X, associates the unique path-connected component $C'$ of Y such that $f(C)\subset C'$. If we identify $\pi_0(X)$ and $\pi_0(Y)$ with [P; X] and [P; Y] respectively, the mapping $\pi_0(f)$ identifies with the mapping $\chi \mapsto [f]\circ \chi$ from [P; X] into [P; Y]. In particular, if $f$ and $g$ are homotopic mappings from X into Y, we have $\pi_0(f) =\pi_0(g)$ (III, p. 230, prop. 2).

Let Z be a topological space and let $g: Y\rightarrow$ Z be a continuous mapping; we have $\pi_0(g\circ f) =\pi_0(g)\circ \pi_0(f)$. In particular, if Z = X and if $f$ and $g$ are homotopy inverses of one another up to homotopy, we have $\pi_0(g)\circ \pi_0(f) =\pi_0$(Id$_X$) $=$ Id$_{\pi_0(X)}$ and $\pi_0(f)\circ \pi_0(g) =\pi_0$(Id$_Y$) $=$ Id$_{\pi_0(Y)}$, which proves that $\pi_0(f)$ and $\pi_0(g)$ are inverse bijections of one another. A space homotopy equivalent to a path-connected space is therefore itself path-connected. In particular, a space homotopy equivalent to a point is path-connected.