Regular open set

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]

As an example, if ${\displaystyle \mathbb {R} }$ has its usual Euclidean topology then every open interval 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. Any degenerate closed interval (meaning an interval of the form ${\displaystyle [x,x]=\{x\}}$ consisting of a single point) 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

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

• 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 (First ed.). Mineola, N.Y.: Dover Publications. ISBN 978-0-486-43479-7. OCLC 115240.