regular closed 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 $\operatorname(\overline) = S$ or, equivalently, if $\partial(\overline)=\partial S,$ where $\operatorname S,$ $\overline$ and $\partial S$ denote, respectively, the interior, closure and boundary of $S.$Steen & Seebach, p. 6

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

Examples

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

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.

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

The interior of a closed subset of $X$ is a regular open subset of $X$ and likewise, the closure of an open subset of $X$ is a regular closed subset of $X.$Willard, "3D, Regularly open and regularly closed sets", p. 29 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 $U \vee V = \operatorname(\overline),$ the meet is $U \and V = U \cap V$ and the complement is $\neg U = \operatorname(X \setminus U).$

See also

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Notes

References

* Lynn Arthur Steen and J. Arthur Seebach, Jr., ''Counterexamples in Topology''. Springer-Verlag, New York, 1978. Reprinted by Dover Publications, New York, 1995. (Dover edition).
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General topology