Exercise 21, § 5

a) Let $(f_n)$ be a sequence of continuous mappings of a topological space $X$ into a perfectly normal space $Y$ (§ 4, Exercise 8) such that, for each $x \in X$, the sequence $(f_n(x))$ has a limit $f(x)$ in $Y$. Show that, for every closed subset $S$ of $Y$, $A = \bar{f}^{-1}(S)$ is a countable intersection of open sets. [There exists a decreasing sequence $(U_n)$ of open sets in $Y$ such that $S = \bigcap_n \overline{U_n}$; note that $f(x) \in S$ if and only if, for each integer $n > 0$, there exists an integer $k \geq 0$ such that $f_{n+k}(x) \in U_n$.] Deduce that $\overline{A} - A$ is meagre in $X$.

b) Deduce from a) that if we suppose in addition that $Y$ is a metrizable space of countable type, then the set of all $x \in X$ such that $f$ is not continuous at $x$ is meagre. [If $(V_n)$ is a base of the topology of $Y$ and if $S_n = Y - V_n$ and $A_n = \bar{f}^{-1}(S_n)$, show that $x$ must belong to one of the sets $\overline{A}_n - A_n$.]