Residual set

Remarks

Equivalently, $R$ contains a set $\bigcap_{n \ge 1} U_n$, where each $U_n$ is open and dense in $X$. Residual sets are also called comeagre sets.

Example: the set of irrational numbers is residual in $\R$. Its complement $\Q$ is countable, so it is a countable union of single points, and each single point is nowhere dense.

Non-example: $\Q$ is not residual in $\R$. Otherwise $\R$ would be the union of the meagre sets $\Q$ and $\R \setminus \Q$. This contradicts the Baire category theorem, since $\R$ is a complete metric space.