Exercise 20, § 2

In the construction of Exercise 23 of Chapter I, § 9, take $X_0$ to be the interval $[0, 1]$ of $\mathbf{R}$. Show that the homeomorphisms of $X$ onto itself are the same as the homeomorphisms of $X_0$ onto itself, although the topology of $X$ is strictly finer than the topology of $X_0$ [use Exercise 20 b) of Chapter I, § 8].