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.^[1]

Properties and examples

Every regular space is semiregular.^[1] The converse is not true. For example, the space ${\displaystyle X=\mathbb {R} ^{2}\cup \{0^{*}\}}$ with the double origin topology^[2] and the Arens square^[3] are Hausdorff semiregular spaces that are not regular.

Open subspaces of a semiregular space are semiregular.^[4] But arbitrary subspaces, even closed subspaces, need not be semiregular.^[4]

The product of an arbitrary family of semiregular spaces is semiregular.^[4]

Every topological space may be embedded into a semiregular space.^[1]

See also

• Separation axiom – Axioms in topology defining notions of "separation"