of a topological space
is called a regular open set if it is equal to the interior of its closure; expressed symbolically, if
denote, respectively, the interior, closure and boundary of
is called a regular closed set if it is equal to the closure of its interior; expressed symbolically, 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.
is a closed subset of
but not a regular closed set because its interior is the empty set
is a regular open set if and only if its complement in
is a regular closed set.
[2] Every regular open set is an open set and every regular closed set is a closed set.
Each clopen subset of
itself) is simultaneously a regular open subset and regular closed subset.
The interior of a closed subset of
is a regular open subset of
and likewise, the closure of an open subset of
is a regular closed subset of
[2] 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.
[2] The collection of all regular open sets in
forms a complete Boolean algebra; the join operation is given by