Definition 2

A subset $A$ of a topological space $X$ is said to be meagre if it is the union of a countable family of nowhere dense sets.

Equivalently, $A$ is meagre if it is contained in a countable union of closed sets each of which has no interior points.

A meagre set can perfectly well be dense in $X$; even the whole space $X$ can itself be a meagre set.

An example of the latter possibility is provided by any countable Hausdorff space with no isolated points, e.g., the rational line $\mathbf{Q}$. But a topological space $X$ which is a meagre set in $X$ need not be countable (see Exercise 9).

Every subset of a meagre set in a space $X$ is meagre, and the union of a countable family of meagre sets is meagre.

Let $Y$ be a subspace of $X$. A subset $A$ of $Y$ is said to be meagre relative to $Y$ if $A$ is meagre when considered as a subset of the topological space $Y$. It follows from Proposition 2 of no. 1 that if $A$ is a subset of $Y$ which is meagre relative to $Y$, then $A$ is meagre relative to $X$; and that if also $Y$ is open in $X$, every subset $A$ of $Y$ which is meagre relative to $X$ is meagre relative to $Y$.