Semiregular space

Definition
A semiregular space is a topological space $$X$$satisfying the following equivalent conditions:


 * 1) The regular open subsets (these are subsets that equal the interior of their closure) form a basis for the space.
 * 2) For any $$x \in X$$ and any open subset $$V \subseteq X$$ containing $$x$$, there exists a regular open subset $$U$$ of $$X$$ containing $$x$$ and contained in $$V$$.

Note that regular open is not the same as being open and regular in the subspace topology. For the notion defined using that, see locally regular space.