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.

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

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