Regular open set

A subset $S$ of a topological space $X$ is called a regular open set if it is equal to the interior of its closure; expressed symbolically, if $\operatorname {Int} ({\overline {S}})=S$ or, equivalently, if $\partial ({\overline {S}})=\partial S,$ where $\operatorname {Int} S,$ ${\overline {S}}$ and $\partial S$ denote, respectively, the interior, closure and boundary of $S.$

A subset $S$ of $X$ is called a regular closed set if it is equal to the closure of its interior; expressed symbolically, if ${\overline {\operatorname {Int} S}}=S$ or, equivalently, if $\partial (\operatorname {Int} S)=\partial S.$