Semiregular space

A semiregular space is a topological space whose regular open sets (sets that equal the interiors of their closures) form a base.

Every regular space is semiregular, and every topological space may be embedded into a semiregular space.

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.