Exercise 14, § 6

(a) Let $X$ be a separated space, $S$ a Souslinian subspace of $X$. Now prove that the set $S$ is approachable in $X$ (IX, p. 69). (Let $P$ be a Polish space, $g$ a continuous mapping of $P$ onto $S$, $(C_n, p_n, \varphi_n)$ a sieve of $P$ and $f$ the corresponding continuous mapping of $L(C)$ onto $P$ (with the notations of IX, p. 65); put $h = g \circ f$, and for every $c \in C_n$, denote by $q_n(c)$ the subspace of $L(C)$ formed by the sequences $(c_k)$ such that $c_n = c$. Let $F_n(c) = h(q_n(c))$, $X_n(c) = g(\varphi_n(c)) \supset F_n(c)$, for every $c \in C_n$; denote by $Z_n(c)$ the union of $D(F_n(c))$ (IX, p. 111, exerc. 3) and of a meagre set in $X$, such that $F_n(c) \subset Z_n(c) \subset \overline{X_n(c)}$, so that $Z_n(c)$ is approachable in $X$. Now prove that

is meagre in $X$, by observing that $F_n(c) = \bigcup_{p_n(c') = c} F_{n+1}(c')$; prove on the other hand that the set $Z_n(c) - F_n(c)$ is contained in the union of the sets $Y_m(d)$, where $m$ runs through the set of integers $\geq n$ and for each $m$, $d$ runs through the set of elements of $C_m$ such that $p_{nm}(d) = c$. Conclude that $F_n(c)$ is approachable). (b) Give an example of a continuous mapping $f$ of $I = \{0, 1\}$ into itself and of an approachable set $B \subset I$ such that $f(B)$ is not approachable. (Use exerc. 16 b) of IV, p. 64, and show that one can take for $B$ a subset of the Cantor set $K$ such that $f(B) = Z$ has the property described in exerc. 17 d) of IX, p. 114).