A subset of a topological space is called a regular open set if it is equal to the interior of its closure; expressed symbolically, if or, equivalently, if where and denote, respectively, the interior, closure and boundary of Steen & Seebach, p. 6
A subset of is called a regular closed set if it is equal to the closure of its interior; expressed symbolically, if or, equivalently, if
Examples
If has its usual Euclidean topology then the open set is not a regular open set, since Every open interval in 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 but not a regular closed set because its interior is the empty set so that
Properties
A subset of is a regular open set if and only if its complement in is a regular closed set. Every regular open set is an open set and every regular closed set is a closed set.
A subset in a topological space is a regular open set if and only if for some Willard, "3D, Regularly open and regularly closed sets", p. 29. This is a consequence of the maximal and minimal properties of the interior and closure operators which when combined, they lead to
Each clopen subset of (which includes and 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 forms a complete Boolean algebra; the join operation is given by the meet is and the complement is
See also
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).