Semiregular space

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

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

The space $$X = \Reals^2 \cup \{0^*\}$$ with the double origin topology and the Arens square are examples of spaces that are Hausdorff semiregular, but not regular.