Regular open set

A subset ${\displaystyle S}${\displaystyle S} of a topological space ${\displaystyle X}${\displaystyle X} is called a regular open set if it is equal to the interior of its closure; expressed symbolically, if ${\displaystyle \operatorname {Int} ({\overline {S}})=S}${\displaystyle \operatorname {Int} ({\overline {S}})=S} or, equivalently, if ${\displaystyle \partial ({\overline {S}})=\partial S,}${\displaystyle \partial ({\overline {S}})=\partial S,} where ${\displaystyle \operatorname {Int} S,}${\displaystyle \operatorname {Int} S,} ${\displaystyle {\overline {S}}}${\displaystyle {\overline {S}}} and ${\displaystyle \partial S}${\displaystyle \partial S} denote, respectively, the interior, closure and boundary of ${\displaystyle S.}${\displaystyle S.}^[1]

A subset ${\displaystyle S}${\displaystyle S} of ${\displaystyle X}${\displaystyle X} is called a regular closed set if it is equal to the closure of its interior; expressed symbolically, if ${\displaystyle {\overline {\operatorname {Int} S}}=S}${\displaystyle {\overline {\operatorname {Int} S}}=S} or, equivalently, if ${\displaystyle \partial (\operatorname {Int} S)=\partial S.}${\displaystyle \partial (\operatorname {Int} S)=\partial S.}^[1]

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

A subset of ${\displaystyle X}${\displaystyle X} is a regular open set if and only if its complement in ${\displaystyle X}${\displaystyle X} is a regular closed set.^[2] Every regular open set is an open set and every regular closed set is a closed set.

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

The interior of a closed subset of ${\displaystyle X}${\displaystyle X} is a regular open subset of ${\displaystyle X}${\displaystyle X} and likewise, the closure of an open subset of ${\displaystyle X}${\displaystyle X} is a regular closed subset of ${\displaystyle X.}${\displaystyle X.}^[2] 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.^[2]

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

• List of topologies – List of concrete topologies and topological spaces
• Regular space – Property of topological space
• Semiregular space
• Separation axiom – Axioms in topology defining notions of "separation"

• Lynn Arthur Steen and J. Arthur Seebach, Jr., Counterexamples in Topology. Springer-Verlag, New York, 1978. Reprinted by Dover Publications, New York, 1995. ISBN 0-486-68735-X (Dover edition).
• Willard, Stephen (2004) [1970]. General Topology. Mineola, N.Y.: Dover Publications. ISBN 978-0-486-43479-7. OCLC 115240.

• General topology