Residual set

A residual set in a topological space $X$ is a subset $R\subseteq X$ whose complement $X\setminus R$ is meager .

Equivalently, $R$ is residual if it contains a countable intersection of sets that are both open and dense . In a Baire space , every residual set is dense.

Examples:

• The irrationals $\mathbb{R}\setminus\mathbb{Q}$ form a residual subset of $\mathbb{R}$, since $\mathbb{Q}$ is meager .
• In any Baire space, the intersection of countably many dense open sets is residual (and dense), as captured by intersection of dense open sets is dense .