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 or, equivalently, if where and \partial S denote, respectively, the interior, closure and boundary of S. A subset S of X is called a regular closed set if it is equal to the closure of its interior; expressed symbolically, if or, equivalently, if

Examples

If \Reals has its usual Euclidean topology then the open set is not a regular open set, since 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 {x} is a closed subset of \R but not a regular closed set because its interior is the empty set so that

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. 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 the meet is and the complement is