Exercise 12, § 6

a) Under the hypotheses of th. 4 of IX, p. 69, let $f$ be a continuous mapping of the quotient space X/R into a metrizable space Y. Prove that $f(Y)$ is a Borel set in Y. b) Let X be a compact metrizable space, Z a nonempty closed subset of X, $\mathfrak{R}$ the set of continuous injective mappings of Z into X. Prove that, if one considers $\mathfrak{R}$ as a subset of the metrizable space $\mathscr{C}_u(Z; X)$ (X, p. 7), $\mathfrak{R}$ is a countable intersection of open sets (consider a countable fundamental system of open neighbourhoods of the diagonal of $X \times X$). c) Equip the set $\mathscr{F}(X)$ of nonempty closed subsets of X with the distance defined in IX, p. 91, exerc. 6; prove that in this metric space, the set $\mathfrak{R}(Z)$ of closed subspaces homeomorphic to Z is Borel. (Consider the mapping $f$ of $\mathfrak{R}$ onto $\mathfrak{R}(Z)$ which, to every mapping $u \in \mathfrak{R}$, associates $u(Z)$; observe that the fibre $f^{-1}(f(u))$ is the set of mappings $u \circ s$, where s runs through the group G of homeomorphisms of Z onto itself.)