Exercise 21, § 2

a) Let $X$ be a metrizable space, let $d$ be a bounded metric compatible with the topology of $X$, let $\sigma$ be the metric corresponding to $d$ on the set $\tilde{\mathfrak{F}}(X)$ of non-empty closed subsets of $X$ (Exercise 6); and let $R$ be an open and closed equivalence relation on $X$. Show that the restriction of $\sigma$ to $X/R$ is a metric compatible with the quotient topology. [Use Exercise 20 b) to show that every class mod $R$ is open and compact. Then prove that if a sequence $(z_n)$ in $X/R$ tends to a point $a$ and if $\varphi : X \to X/R$ is the canonical mapping, we have $\sigma(\overline{\varphi^{-1}(a)}, \overline{\varphi^{-1}(z_n)}) \to 0$; argue by contradiction, using the fact that $R$ is open.]

b) On the compact interval $I = [0, 1]$ of $\mathbf{R}$, let $R$ be the equivalence relation whose classes are the points of the Cantor set $K$ (Chapter VI, § 2, no. 5), other than the end-points of the intervals contiguous to $K$, and the closures of the intervals contiguous to $K$. Show that the quotient space $I/R$ is homeomorphic to $I$ [Chapter IV, § 8, Exercise 16 b)] but that the metric $\sigma$ is not compatible with the topology of $I/R$.