Topological basis with some closure properties

Question

Given a topological space $(X,\tau)$, I'm interested in a topological basis $\mathcal{B}$ for $\tau$, with the following property:

For every open subset $U\neq\emptyset$ of $X$ and for every $x \in U$ there exists a non-empty $V\in\mathcal{B}$ such that $x \in V$ and $\text{Cl}(V)\subseteq U$

Is there a name for this stronger basis? Is it a known property?
Thanks!

Answer 1

Any base on a regular space has this property. Having such a base implies regularity as well. So it's equivalent to $X$ being regular.