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 $$\partial(\operatorname{Int}S)=\partial S.$$

Examples
If $$\Reals$$ has its usual Euclidean topology then the open set $$S = (0,1) \cup (1,2)$$ is not a regular open set, since $$\operatorname{Int}(\overline{S}) = (0,2) \neq S.$$ Every open interval in $$\R$$ 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 $$\{x\}$$ is a closed subset of $$\R$$ but not a regular closed set because its interior is the empty set $$\varnothing,$$ so that $$\overline{\operatorname{Int} \{x\}} = \overline{\varnothing} = \varnothing \neq \{x\}.$$

Properties
A subset of $$X$$ is a regular open set if and only if its complement in $$X$$ is a regular closed set. Every regular open set is an open set and every regular closed set is a closed set.

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

The interior of a closed subset of $$X$$ is a regular open subset of $$X$$ and likewise, the closure of an open subset of $$X$$ is a regular closed subset of $$X.$$ 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.

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