# Open sets containing closed set which contain open sets

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?

• That is the exact definition of a regular topological space. – bof Apr 19 '18 at 5:40
• Ah, I see; the construction I was thinking about actually requires compactness (! which I didn't state in the question, so it is besides the point), but yes, thank you! – Guillermo Angeris Apr 19 '18 at 5:49

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 $x\in H\subseteq K\subseteq G$.