Regular open set

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

A subset <math>S</math> of <math>X</math> is called a regular closed set if it is equal to the closure of its interior; expressed symbolically, if <math>\overline{\operatorname{Int} S} = S</math> or, equivalently, if <math>\partial(\operatorname{Int}S)=\partial S.</math>. This is a consequence of the maximal and minimal properties of the interior and closure operators which when combined, they lead to

<math>\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}</math>

<math>\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}</math>

Each clopen subset of <math>X</math> (which includes <math>\varnothing</math> and <math>X</math> 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.