Baire space

Definition

A topological space $X$ is a Baire space when it has the property that follows. Let $(F_n)_{n \in \mathbb{N}}$ be a countable family of closed subsets of $X$ with $\operatorname{int} F_n = \emptyset$ for each $n$. Then also $\operatorname{int} \bigcup_n F_n = \emptyset$.

Remarks

Here $\operatorname{int}$ is the interior. Equivalently, in a Baire space each countable intersection of dense open sets is dense. In a Baire space, each meagre set has empty interior. Thus a nonempty Baire space is not meagre in itself. Each complete metric space is a Baire space by the Baire category theorem. The space $\mathbb{Q}$ is not a Baire space.

Explanations

L1 human

In a Baire space, countably many thin closed sets can never fill up an open set. This says that the space is large. It lets you prove that some objects exist: you show that the bad cases form a small set.