Exercise 10, § 6

a) Let X be a metrizable space and let S be a Souslin subspace of X. Show that S is almost open in X (§ 5, Exercise 6). [Let P be a Polish space, g a continuous mapping of P onto S, $(C_n, p_n, \varphi_n)$ a sifting of P, and f the corresponding continuous mapping of L(C) onto P (in the notation of no. 5); let $h = g \circ f$, and for each $c \in C_n$ let $q_n(c)$ denote the subspace of L(C) consisting of sequences $(c_k)$ such that $c_n = c$. Let $F_n(c) = h(q_n(c))$ and let $U_n(c) = g(\varphi_n(c)) \supset F_n(c)$ for each $c \in C_n$; let $W_n(c)$ denote the union of $D(U_n(c))$ (§ 5, Exercise 3) and a meagre set in X, such that $F_n(c) \subset W_n(c) \subset \overline{U_n(c)}$, so that $W_n(c)$ is almost open in X. Show that

is meagre in X by noting that $F_n(c) = \bigcup_{p_n(c') = c} F_{n+1}(c')$; show also that the set $W_n(c) - F_n(c)$ is contained in the union of the sets $V_m(d)$ where $m \geq n$ and, for each $m, d$ runs through the set of elements of $C_m$ such that $p_{nm}(d) = c$. Hence show that $F_n(c)$ is almost open.]

b) Give an example of a continuous mapping $f$ of $[0, 1]$ into itself and an almost open set $B \subset I$ such that $f(B)$ is not almost open. [Use Exercise 16 b) of Chapter IV, § 8, and show that we may take $B$ to be a subset of the Cantor set $K$ such that $f(B) = Z$ has the property described in § 5, Exercise 18 d).]