of a topological space is called a regular open set if it is equal to the interior of its closure denote, respectively, the interior, closure and boundary

is called a regular closed set

Examples

has its usual Euclidean topology Every open interval

Properties

is a regular open set if and only if is a regular closed set. Every regular open set is an open set and every regular closed set is a closed set.

. 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 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.

forms a complete Boolean algebra; the join the meet

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).