This is perhaps a more general question, but is there any name for a topological space which supports the following structure? Given any open set $A$ in the topology and some point $x$, there exists a closed set $B$ and another open set $A'$ which satisfy
$$ x \in A' \subset B \subset A $$ ?
That such a structure is supported is clear in, e.g., metric spaces (with the usual topology generated by the distance function) by letting $S$ be the open ball which contains $x$ and picking $B$ to be the closed ball centered at $x$ with half of the distance of $x$ to the boundary of $A$. Then $A'$ can simply be, say, the interior of $B$.
Perhaps there may not be a name for this particular case, but is there a name for an equivalent (or slightly stronger) property of a space?
An Example:
Locally compact Hausdorff space $X$: For every $x\in X$ and any open set $G$ that containing $x$, there is a compact set $K$ and an open set $H$ such that $x\in H\subseteq K\subseteq G$.