here's a problem from Dugundji's book:
Let $X$ be Hausdorff and assume that each $x \in X$ has a nbd $V$ such that $\overline{V}$ is regular. Prove $X$ is regular.
Recall that $X$ is regular iff every $x \in X$ has a local base of closed nbds. So here's what I tried:
Let $x$ be in $X$ and let $W$ be any open set containing $x$. By assumption $x$ has a nbd V such that $\overline{V}$ is regular. This means $\overline{V}$ has a base consisting of closed neighborhoods.
Note $W \cap \overline{V}$ is open in $\overline{V}$ so we may find a closed set $C$ and an open set $U$ such that:
$x \in U \subset C \subset W \cap \overline{V}$.
Since $C$ is closed then $\overline{U} \subset C$.
Hence $x \in U \subset C \subset \overline{W} \subset W \cap \overline{V} \subset W$.
Thus $X$ is regular. Is this OK, if not, how to prove it? thanks in advance!