The question also can be seen here.
Let $X$ be a Moore space and $e(X)=\omega$. Is it metrizable?
What I've tried: I list these facts:
1 A space $X$ is a Moore space iff $X$ is a $\sigma$-space and a $p$-space.
2 If $X$ is a $p$-space, then $nw(X)=w(X)$.
3 A space $X$ is a $\sigma$-space if $X$ has a $\sigma$-discrete network.
Since if $X$ is a $\sigma$-space and $e(X)=\omega$, we know that $nw(X)=\omega$, see the proof. So by 2, we can conclude that $w(X)=nw(X)=\omega$, and hence it is metrizable.
However I'm not sure. Thanks for any help.