Countable sets have measure zero. Given the analogies between sets of measure zero and meagre sets (both modelling some idea of "smallness"), are countable sets meagre?
If this is true in some spaces but not others, I would be particularly interested to find out if countable sets are meagre in the Cantor space.
In a $T_1$ space without isolated points (such spaces are often called dense in themselves or "crowded") this holds, as then a countable set $D$ is just $\bigcup \{\{d\} : d \in D\}$ and each set $\{d\}$ is nowhere dense (closed by $T_1$ and its closure, i.e. $\{d\}$, has empty interior as $d$ is not isolated in $X$). And meagre sets are just countable unions of nowhere dense sets by definition.
The Cantor set is of course just such a space.
If you're interested in these topics, check out the book "Measure and Category" by Oxtoby, a nicely self-contained exploration of the analogues between things like sets of measure $0$ and meagre sets. It starts with the classic example of writing $\mathbb{R} = M \cup N$ where $N$ is a null set (has Lebesgue measure $0$) and $M$ is meagre. To see how different such "small" sets can be.