Exercise 9, § 6

a) Let X be a Polish space, and let R be an equivalence relation on X. Show that there exists a subset of X which is a countable inter-

() \ It can be shown that in the space $C(I; \mathbf{R})$ of continuous real-valued functions on a compact interval $I \subset \mathbf{R}$, endowed with the topology of uniform convergence (which makes it into a Polish space), the set of differentiable functions is not a Souslin set but its complement is a Souslin set [cf. S. Mazurkiewicz, Fund. Math. 27 (1936), p. 244]. \* section of open sets and which meets each equivalence class in exactly one point, provided one or the other of the following two hypotheses is satisfied:

α) R is closed; β) X is locally compact and the graph of R is closed in $X \times X$.

[For α), follow the method of the proof of Theorem 4 of no. 8, making use of the following remark: If $(G_n)$ is a sequence of sets, each of which is a countable intersection of open sets, and if for each $n$ there exists a neighbourhood $V_n$ of $G_n$ such that the family $(V_n)$ is locally finite, then $\bigcup_n G_n$ is a countable intersection of open sets. Same method for β); observe that the saturation of a compact set is now closed (cf. Chapter I, § 10, Exercise 16).]

b) Let X be a Polish space, let R be an equivalence relation on X which is both open and closed, and let $\varphi : X \to X/R$ be the canonical mapping. Show that there exists a subset A of X which is a countable intersection of open sets in X and is such that the restriction of $\varphi$ to A is a homeomorphism of A onto $\varphi(A)$ and such that $\varphi(A)$ is dense in $X/R$ and is a countable intersection of open sets. [With the notation of the proof of Theorem 4 of no. 8, show that we may assume that for each $n$ the $\varphi_n(c)$ have been defined in such a way that the image under $\varphi$ of the union of the interiors of the $h_n(c)$, as $c$ runs through $C_n$, is dense in $X/R$; use a) and Theorem 1 of no. 1.]

c) Show that the conclusion of b) remains valid when X is locally compact and has a countable base and R is a closed equivalence relation on X all of whose equivalence classes are compact. [Reduce to case b) by the use of § 5, Exercise 26.]