Definition 3

A topological space $X$ is said to be a Baire space if it satisfies one or the other of the following two equivalent conditions:

(EB) Every countable intersection of dense open sets in $X$ is dense in $X$. (EB') Every countable union of closed sets with no interior points in $X$ has no interior point in $X$.

Axiom (EB) can be stated in two other equivalent forms:

(EB'') Every non-empty open set in $X$ is non-meagre.

Indeed a set is meagre if and only if it is contained in a countable union of closed sets with no interior points.

(EB'') The complement of a meagre set in $X$ is dense in $X$.

This signifies that a meagre set cannot contain a non-empty open set, and is therefore equivalent to (EB").