Exercise 5, § 5

a) Let $X$ and $Y$ be two topological spaces and let $A$ be a nowhere dense subset of $X \times Y$. If the topology of $Y$ has a countable base, show that the set of all $x \in X$ such that the section $A \cap \{x\} / Y$ of $A$ at $x$ is not nowhere dense relative to $\{x\} \times Y$ is a meagre set in $X$ (if $U$ is any non-empty open set in $Y$, show that the set of all $x \in X$ such that $\{x\} \times U$ is contained in the closure of the section of $A$ at $x$ is a nowhere dense set in $X$).

b) Show by an example that the result of a) can be false if the topology of $Y$ does not have a countable base (take $X$ to be a Hausdorff space with no isolated points and $Y$ to be the set $X$ with the discrete topology; and take $A$ to be the diagonal of $X \times Y$).

c) If $B, C$ are subsets of $X, Y$ respectively and if one of $B, C$ is meagre (in $X, Y$ respectively), then $B \times C$ is meagre in $X \times Y$. Conversely, if $B \times C$ is meagre in $X \times Y$ and if the topology of either $X$ or $Y$ has a countable base, then one of the sets $B, C$ is meagre.

d) Deduce from c) that if the topology of one of the spaces $X, Y$ has a countable base, then $D(B \times C) = D(B) \wedge D(C)$ in the notation of Exercise 3.