In the book by Oxtoby, the chapter 16 is devoted to them and Banach Category Theorem. However I do not understand this chapter and parts of the following one.
I know that I can define any topology I want, for example $2^X$, the discrete topology. And then every singleton is open. But is it of first category? Are there any nowhere-dense subset there? I don't think so.
On the other hand, in metric space, there is a ball around every point in open set and I think it can't be of second category, surely not for complete spaces.
I think I need another topology, where such things are possible, to understand what is Banach Category Theorem about.
Every open set in the metric space $\Bbb Q$ (with the usual Euclidean metric) is meagre (i.e., first category) in $\Bbb Q$. In fact every subset of $\Bbb Q$ is meagre in $\Bbb Q$, because every subset of $\Bbb Q$ is countable, and $\{x\}$ is a closed, nowhere dense subset of $\Bbb Q$ for each $x\in\Bbb Q$.
For a slightly more interesting example, let $C$ be the middle-thirds Cantor set, and let $X=\Bbb Q\times C$ with the topology that it inherits from $\Bbb R^2$. For each $x\in\Bbb Q$ the set $\{x\}\times C$ is closed and nowhere dense in $X$, so $(U\times V)\cap X$ is a meagre open set in $X$ whenever $U$ and $V$ are open sets in $\Bbb R$ and $[0,1]$, respectively.