TITLE: A locally metrizable space
QUESTION [6 upvotes]: Is every locally metrizable space always first countable? 
A locally metrizable space $X$ means for every point $x\in X$ has an open nbhd such that it is metrizable.
Thanks for help.

REPLY [3 votes]: Yes.
A topological space is called first countable if every point has a countable neighborhood basis.
Fix $x\in X$ arbitrary.  Fix $U\subseteq X$ a neighborhood of $x$ which is metrizable, say by $\rho:U\times U\to[0,+\infty]$ (or $[0,+\infty)$), so $(U,\rho)$ is a metric space.  You don't even need to do things like $B(x,1/n)\cap U$ because the metric isn't defined off of $U\times U$.
Then as the original topology and the topology induced by $\rho$ are identical, what Damien L said is true.  $\{B(x,1/n)\}_{n\in\mathbb{N}}$ is the countable neighborhood basis we need.