Example 3

One says that a pointed topological space $(X, x)$ is contractible, or that the topological space X is contractible at $x$, if there exists a homotopy pointed at $x$ connecting Id$_X$ to the constant mapping of X into X of image $\{x\}$. Such a space is homotopy equivalent to a point. There nevertheless exist spaces homotopy equivalent to a point which are not contractible at any of their points, and spaces contractible at one point, but not at every point (III, p. 321, exerc. 1).