Regular open set

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

Examples

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

Properties

A subset of [math]\displaystyle{ X }[/math] is a regular open set if and only if its complement in [math]\displaystyle{ X }[/math] 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 [math]\displaystyle{ X }[/math] (which includes [math]\displaystyle{ \varnothing }[/math] and [math]\displaystyle{ X }[/math] itself) is simultaneously a regular open subset and regular closed subset.

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

See also

• List of topologies – List of concrete topologies and topological spaces
• Regular 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). General Topology. Dover Books on Mathematics (First ed.). Mineola, N.Y.: Dover Publications. ISBN 978-0-486-43479-7. OCLC 115240.