Editing Open set (section)

=== Regular open sets{{anchor|Regular open set|Regular closed set}} ===

A subset <math>S</math> of a topological space <math>X</math> is called a '''{{em|[[regular open set]]}}''' if <math>\operatorname{Int} \left( \overline{S} \right) = S</math> or equivalently, if <math>\operatorname{Bd} \left( \overline{S} \right) = \operatorname{Bd} S</math>, where <math>\operatorname{Bd} S</math>, <math>\operatorname{Int} S</math>, and <math>\overline{S}</math> denote, respectively, the topological [[Boundary (topology)|boundary]], [[Interior (topology)|interior]], and [[Closure (topology)|closure]] of <math>S</math> in <math>X</math>.  A topological space for which there exists a [[Base (topology)|base]] consisting of regular open sets is called a '''{{em|[[semiregular space]]}}'''. 
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, where by definition a subset <math>S</math> of <math>X</math> is called a '''{{em|[[regular closed set]]}}''' if <math>\overline{\operatorname{Int} S} = S</math> or equivalently, if <math>\operatorname{Bd} \left( \operatorname{Int} S \right) = \operatorname{Bd} S.</math> 
Every regular open set (resp. regular closed set) is an open subset (resp. is a closed subset) although in general,<ref group=note>One exception if the if <math>X</math> is endowed with the [[discrete topology]], in which case every subset of <math>X</math> is both a regular open subset and a regular closed subset of <math>X.</math></ref> the converses are {{em|not}} true.