Let $X$ be a Moore space and $e(X)=\omega$. Is it metrizable?

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.

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