Locally regular space

Definition
A topological space $$X$$ is termed locally regular if it satisfies the following equivalent conditions:


 * 1) It has a basis of open subsets each of which is a regular space under the subspace topology.
 * 2) For any $$x \in X$$, there exists an open subset $$U$$ containing $$x$$ such that $$U$$ is a regular space with the subspace topology.
 * 3) For any $$x \in X$$ and open subset $$V$$ containing $$x$$, there exists an open subset $$U$$ containing $$x$$ such that $$U \subseteq V$$ and $$U$$ is a regular space with the subspace topology.