Exercise 12, § 5

A topological space $X$ is said to be totally non-meagre if every non-empty closed subspace of $X$ is non-meagre. A locally compact space is totally non-meagre, and so is a complete metric space.

a) Show that a totally non-meagre regular space is a Baire space.

b) In an accessible totally non-meagre space, a non-empty countable closed set is thin (Exercise 2) and hence has at least one isolated point.

c) In a totally non-meagre space $X$, every subspace $A$ which is the intersection of a countable intersection of open sets in $X$ is totally non-meagre (if $F$ is a subset of $A$ which is closed in $A$, let $\overline{F}$ be the closure of $F$ in $X$, and show that $\overline{F} \cap C_F$ is meagre relative to $\overline{F}$).

d) If every point of a topological space $X$ has a neighbourhood which is a totally non-meagre subspace, then $X$ is totally non-meagre.