Meagre set

Definition

Let $X$ be a topological space and let $A \subseteq X$. The set $A$ is meagre in $X$ when there is a countable family $(A_n)_{n \in \mathbb{N}}$ of nowhere dense subsets of $X$ with $A = \bigcup_{n} A_n$.

Remarks

Another name for a meagre set is a set of first category. A set that is not meagre is of second category. Each subset of a meagre set is meagre, and each countable union of meagre sets is meagre. The set $\mathbb{Q}$ is meagre in $\mathbb{R}$, because it is a countable union of one-point sets. By the Baire category theorem, $\mathbb{R}$ itself is not meagre in $\mathbb{R}$.

Explanations

L1 human

A meagre set is made of countably many thin pieces. It can still be dense, as $\mathbb{Q}$ is, but in a complete metric space it can never fill an open set.