Meager set

A meager set in a topological space $X$ is a subset $M\subseteq X$ that can be written as a countable union of nowhere dense sets .

Meager sets are also called “sets of first category.” Their complements are the residual sets ; in a Baire space , meager sets are topologically “small.”

Examples:

• Any countable subset of $\mathbb{R}$ (with the usual topology), such as the rationals $\mathbb{Q}$, is meager (a countable union of singleton sets, each nowhere dense).
• Any nowhere dense set is meager (as a countable union of itself).