Residual set

Let $(X,d)$ be a metric space . A set $R\subseteq X$ is residual (or comeager) if its complement is meager : $X\setminus R\ \text{is meager in }X.$

Residual sets are “topologically large” in complete metric spaces : by the Baire category theorem (see Baire space ), residual sets are dense (in fact, they contain a dense $G_\delta$ set, though that terminology requires additional definitions).

Examples:

• In $\mathbb{R}$, the set of irrational numbers $\mathbb{R}\setminus\mathbb{Q}$ is residual, since $\mathbb{Q}$ is meager.
• If $X$ is a complete metric space, then $X$ itself is residual in $X$.
• The complement of a nowhere dense set need not be residual unless the set is meager; residualness is a “countable” notion.