Regular open set

Remarks

A regular open set is open, since an interior is always open. For every open $U$, one has $U \subseteq \operatorname{int} \overline{U}$, and the set $\operatorname{int} \overline{U}$ is regular open.

Example: in $\R$, let $U = (0, 1)$. Then $\overline{U} = [0, 1]$ and $\operatorname{int} [0, 1] = (0, 1) = U$, so $U$ is regular open.

Non-example: in $\R$, let $U = (0, 1) \cup (1, 2)$. Then $\overline{U} = [0, 2]$ and $\operatorname{int} \overline{U} = (0, 2) \ne U$, so $U$ is open but not regular open.

Dually, a set $F$ is regular closed if $F = \overline{\operatorname{int} F}$. A set is regular open if and only if its complement is regular closed.