Meager set

Let $(X,d)$ be a metric space . A set $M\subseteq X$ is meager (or of first category) if there exist nowhere dense sets $A_1,A_2,\dots\subseteq X$ such that $M=\bigcup_{n=1}^\infty A_n.$

Meager sets are “topologically small.” The Baire category theorem says that complete metric spaces cannot be meager in themselves, which yields strong “generic” existence statements.

Examples:

• Any countable subset of $\mathbb{R}$ is meager, since a singleton $\{x\}$ is nowhere dense and a countable set is a countable union of singletons.
• $\mathbb{Q}$ is meager in $\mathbb{R}$ (it is countable).
• A meager set can be dense: $\mathbb{Q}$ is dense but meager in $\mathbb{R}$.