Exercise 24, § 5

Let $X,\ Y$ be two topological spaces. A mapping $f : X \to Y$ is said to be almost open if, for every closed subset $S$ of $Y$, $\overline{f^{-1}(S)}$ is almost open in $X$ (Exercise 6).

a) Let $g : X \to Y$ be a mapping with the property that there exists a meagre set $M$ in $X$ such that $g|_M$ is continuous. Show that $g$ is almost open. Consider the converse, when the topology of $Y$ has a countable base.

b) Suppose that $Y$ is perfectly normal ($\S$ 4, Exercise 8). Let $(f_n)$ be a sequence of almost open mappings of $X$ into $Y$ such that

exists for each $x \in X$. Show that $f$ is almost open [argue as in Exercise 21 a)].