Semiregular space

A semiregular space is a topological space whose regular open sets (sets that equal the interiors of their closures) form a base.
Semiregular spaces should not be confused with locally regular spaces, spaces in which there is a base of open sets that induce regular subspaces. For example, the bug-eyed line is locally regular but not semiregular. 
Definitions.
Regular open and regular closed sets.
A subset formula_1 of a topological space formula_2 is called a if formula_3 or equivalently, if formula_4 where formula_5 (resp. formula_6 formula_7) denotes the topological boundary (resp. interior, closure) of formula_1 in formula_9 
A subset of formula_2 is a regular open set if and only if its complement in formula_2 is a regular closed set, where by definition a subset formula_1 of formula_2 is called a if formula_14 or equivalently, if formula_15 
Every regular open set is necessarily an open set and every regular closed set is necessarily a closed set, although in general, the converses are necessarily true. 
Each of formula_2 and formula_17 is simultaneously a regular open subset and regular closed subset of formula_9 
The interior (taken in formula_2) of any closed subset of formula_2 is necessarily a regular open subset of formula_2 and likewise, the closure (taken in formula_2) of any open subset of formula_2 is necessarily a regular closed subset of formula_9 The intersection (although not necessarily the union) of two regular open sets is once again a regular open set. Similarly, the union (although not necessarily the intersection) of two regular closed sets is once again a regular closed set.
If formula_25 has its usual Euclidean topology then every open interval is a regular open subset and every non-degenerate closed interval (that is, a closed interval containing at least two distinct points) is a regular closed subset. Any degenerate closed interval (meaning an interval of the form formula_26 which consists of only a single point) is a closed subset of formula_25 but a regular closed subset because its interior is the empty set formula_17 so that 
Semiregular spaces.
A topological space for which there exists a base consisting of regular open sets is called a . Equivalently, it is any topological space for which the set of all regular open subsets forms a base.
Examples and sufficient conditions.
Every regular space is semiregular, and every topological space may be embedded into a semiregular space. 