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.