Regular open set
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
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 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 thatProperties
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. 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 is simultaneously a regular open subset and regular closed subset.
The intersection of two regular open sets is a regular open set. Similarly, the union 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