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.[1]
Every regular space is semiregular, and every topological space may be embedded into a semiregular space.[1]
The space with the double origin topology[2] and the Arens square[3] are examples of spaces that are Hausdorff semiregular, but not regular.