Exercise 6, § 5

A set $A$ in a topological space $X$ is said to be almost open if there is an open set $U$ in $X$ such that both $U \cap \complement A$ and $A \cap \complement U$ are meagre sets.

a) The complement of an almost open set is almost open.

b) Any countable union of almost open sets is almost open.

c) Show that the following statements are equivalent: $\alpha)$ $A$ is almost open; $\beta)$ there is a meagre subset $M$ of $X$ such that $A \cap \complement M$ is both open and closed in the subspace $X - M$; $\gamma)$ there is a set $G \subset A$ which is a countable intersection of open sets in $X$, such that $A - G$ is meagre in $X$; $\delta)$ there is a set $F \supset A$ which is a countable union of closed sets in $X$, such that $F - A$ is meagre in $X$; $\zeta)$ the set $D(A) \cap D(X - A)$ (Exercise 3) is nowhere dense in $X$; $\theta)$ the set $D(A) \cap \complement A$ is meagre in $X$. [Use Exercises 3 a) and 3 b) to show that $\alpha) \rightarrow \zeta) \rightarrow \theta) \rightarrow \alpha)$.]

d) Let $X, Y$ be two topological spaces and suppose that the topology of either $X$ or $Y$ has a countable base. Then a subset of $X / Y$ of the form $A \times B$ is almost open if and only if either both $A$ and $B$ are almost open or one of them is meagre [use $\zeta$ of $c$].