Example 2

Let X be a nonempty topological space. In order that X be homotopy equivalent to a space reduced to one point, it is necessary and sufficient that there exist a constant mapping $p: X\rightarrow$ X which is homotopic to the identity mapping of X. (This is then so for every constant mapping $q: X\rightarrow X$, since $[p] = [p]\circ [q] =$ [Id$_X]\circ [q] = [q]$.) Indeed, let P be a space reduced to one point, $f: P\rightarrow X$ a mapping and $g: X\rightarrow P$ the unique mapping of X into P. We have $g\circ f=$ Id$_P$; in order that $f$ be a homotopy equivalence, it is necessary and sufficient that $f\circ g$ be homotopic to Id$_X$. Now, $f\circ g$ is constant, of image $f(P)$.