Regular open set

A subset ${\displaystyle S}$ of a topological space ${\displaystyle X}$ is called a regular open set if it is equal to the interior of its closure; expressed symbolically, if ${\displaystyle \operatorname {Int} ({\overline {S}})=S}$ or, equivalently, if ${\displaystyle \partial ({\overline {S}})=\partial S,}$ where ${\displaystyle \operatorname {Int} S,}$ ${\displaystyle {\overline {S}}}$ and ${\displaystyle \partial S}$ denote, respectively, the interior, closure and boundary of ${\displaystyle S.}$¹

A subset ${\displaystyle S}$ of ${\displaystyle X}$ is called a regular closed set if it is equal to the closure of its interior; expressed symbolically, if ${\displaystyle {\overline {\operatorname {Int} S}}=S}$ or, equivalently, if ${\displaystyle \partial (\operatorname {Int} S)=\partial S.}$¹

If ${\displaystyle \mathbb {R} }$ has its usual Euclidean topology then the open set ${\displaystyle S=(0,1)\cup (1,2)}$ is not a regular open set, since ${\displaystyle \operatorname {Int} ({\overline {S}})=(0,2)\neq S.}$ Every open interval in ${\displaystyle \mathbb {R} }$ is a regular open set and every non-degenerate closed interval (that is, a closed interval containing at least two distinct points) is a regular closed set. A singleton ${\displaystyle \{x\}}$ is a closed subset of ${\displaystyle \mathbb {R} }$ but not a regular closed set because its interior is the empty set ${\displaystyle \varnothing ,}$ so that ${\displaystyle {\overline {\operatorname {Int} \{x\}}}={\overline {\varnothing }}=\varnothing \neq \{x\}.}$

A subset of ${\displaystyle X}$ is a regular open set if and only if its complement in ${\displaystyle X}$ is a regular closed set.² Every regular open set is an open set and every regular closed set is a closed set.

A subset ${\displaystyle G}$ in a topological space ${\displaystyle X}$ is a regular open set if and only if ${\displaystyle G=\operatorname {Int} ({\overline {A}})}$ for some ${\displaystyle A\subset X}$². This is a consequence of the maximal and minimal properties of the interior and closure operators which when combined, they lead to

${\displaystyle {\begin{aligned}\operatorname {Int} ({\overline {A}})\subset {\overline {\operatorname {Int} ({\overline {A}})}}\quad \Longrightarrow \quad \operatorname {Int} ({\overline {A}})\subset \operatorname {Int} {\Big (}{\overline {\operatorname {Int} ({\overline {A}})}}{\Big )}\end{aligned}}}$

${\displaystyle {\begin{aligned}\operatorname {Int} ({\overline {A}})\subset {\overline {A}}\quad \Longrightarrow \quad {\overline {\operatorname {Int} ({\overline {A}})}}\subset {\overline {A}}\quad \Longrightarrow \quad \operatorname {Int} {\Big (}{\overline {\operatorname {Int} ({\overline {A}})}}{\Big )}\subset \operatorname {Int} ({\overline {A}})\end{aligned}}}$

Each clopen subset of ${\displaystyle X}$ (which includes ${\displaystyle \varnothing }$ and ${\displaystyle X}$ itself) is simultaneously a regular open subset and regular closed subset.

The intersection (but not necessarily the union) of two regular open sets is a regular open set. Similarly, the union (but not necessarily the intersection) of two regular closed sets is a regular closed set.²

The collection of all regular open sets in ${\displaystyle X}$ forms a complete Boolean algebra; the join operation is given by ${\displaystyle U\vee V=\operatorname {Int} ({\overline {U\cup V}}),}$ the meet is ${\displaystyle U\land V=U\cap V}$ and the complement is ${\displaystyle \neg U=\operatorname {Int} (X\setminus U).}$