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

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

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

The interior of a closed subset of ${\displaystyle X}$  is a regular open subset of ${\displaystyle X}$  and likewise, the closure of an open subset of ${\displaystyle X}$  is a regular closed subset of ${\displaystyle X.}$ ^[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 ${\displaystyle X}$  forms a complete Boolean algebra; the join operation is given by ${\displaystyle U\vee V=\operatorname {Int} ({\overline {U\cup V}}),}$  the meet is ${\displaystyle U\land V=U\cap V}$  and the complement is ${\displaystyle \neg U=\operatorname {Int} (X\setminus U).}$ 

• List of topologies – List of concrete topologies and topological spaces
• Regular space – topological space in which a point and a closed set are, if disjoint, separable by neighborhoodsPages displaying wikidata descriptions as a fallback
• 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.