Editing Regular open set (section)

==Properties==

A subset of <math>X</math> is a regular open set if and only if its complement in <math>X</math> is a regular closed set.<ref name="willard-regopen"/>  Every regular open set is an [[open set]] and every regular closed set is a [[closed set]].

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 interior of a closed subset of <math>X</math> is a regular open subset of <math>X</math> and likewise, the closure of an open subset of <math>X</math> is a regular closed subset of <math>X.</math><ref name="willard-regopen">Willard, "3D, Regularly open and regularly closed sets", p. 29</ref>  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.<ref name="willard-regopen"/>

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