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 $x\in X$ 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.

Relation with other properties

Stronger properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
regular space