Semiregular space

A subset ${\displaystyle S}$ of a topological space ${\displaystyle X}$ is called a regular open set if ${\displaystyle \operatorname {Int} \left({\overline {S}}\right)=S}$ or equivalently, if ${\displaystyle \operatorname {Bd} \left({\overline {S}}\right)=\operatorname {Bd} S,}$ where ${\displaystyle \operatorname {Bd} S}$ (resp. ${\displaystyle \operatorname {Int} S,}$ ${\displaystyle {\overline {S}}}$) denotes the topological boundary (resp. interior, closure) of ${\displaystyle S}$ in ${\displaystyle X.}$ A subset of ${\displaystyle X}$ is a regular open set if and only if its complement in ${\displaystyle X}$ is a regular closed set, where by definition a subset ${\displaystyle S}$ of ${\displaystyle X}$ is called a regular closed set if ${\displaystyle {\overline {\operatorname {Int} S}}=S}$ or equivalently, if ${\displaystyle \operatorname {Bd} \left(\operatorname {Int} S\right)=\operatorname {Bd} S.}$ Every regular open set is necessarily an open set and every regular closed set is necessarily a closed set, although in general,[note 1] the converses are not necessarily true.

Each of ${\displaystyle X}$ and ${\displaystyle \varnothing }$ is simultaneously a regular open subset and regular closed subset of ${\displaystyle X.}$ The interior (taken in ${\displaystyle X}$) of any closed subset of ${\displaystyle X}$ is necessarily a regular open subset of ${\displaystyle X}$ and likewise, the closure (taken in ${\displaystyle X}$) of any open subset of ${\displaystyle X}$ is necessarily a regular closed subset of ${\displaystyle X.}$ The intersection (although not necessarily the union) of two regular open sets is a regular open set. Similarly, the union (although not necessarily the intersection) of two regular closed sets is a regular closed set.

If ${\displaystyle \mathbb {R} }$ has its usual Euclidean topology then every open interval is a regular open subset and every non-degenerate closed interval (that is, a closed interval containing at least two distinct points) is a regular closed subset. Any degenerate closed interval (meaning an interval of the form ${\displaystyle [x,x]=\{x\},}$ which consists of only a single point) is a closed subset of ${\displaystyle \mathbb {R} }$ but not a regular closed subset because its interior is the empty set ${\displaystyle \varnothing }$ so that

${\displaystyle {\overline {\operatorname {Int} \{x\}}}={\overline {\varnothing }}=\varnothing \neq \{x\}.}$

A topological space for which there exists a base consisting of regular open sets is called a semiregular space. Equivalently, it is any topological space for which the set of all regular open subsets forms a base.