Regular open set

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

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

Examples

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

Properties

A subset of $X$ is a regular open set if and only if its complement in $X$ is a regular closed set. Every regular open set is an open set and every regular closed set is a closed set.

A subset $G$ in a topological space $X$ is a regular open set if and only if $G=\operatorname {Int} ({\overline {A}})$ for some $A\subset X$. This is a consequence of the maximal and minimal properties of the interior and closure operators which when combined, they lead to

${\begin{aligned}\operatorname {Int} ({\overline {A}})\subset {\overline {\operatorname {Int} ({\overline {A}})}}\quad \Longrightarrow \quad \operatorname {Int} ({\overline {A}})\subset \operatorname {Int} {\Big (}{\overline {\operatorname {Int} ({\overline {A}})}}{\Big )}\end{aligned}}$

${\begin{aligned}\operatorname {Int} ({\overline {A}})\subset {\overline {A}}\quad \Longrightarrow \quad {\overline {\operatorname {Int} ({\overline {A}})}}\subset {\overline {A}}\quad \Longrightarrow \quad \operatorname {Int} {\Big (}{\overline {\operatorname {Int} ({\overline {A}})}}{\Big )}\subset \operatorname {Int} ({\overline {A}})\end{aligned}}$

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

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.

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

See also

• Regular space – Property of topological space
• Semiregular space
• Separation axiom – Axioms in topology defining notions of "separation"

References

• 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.