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 ...
The question is also posted here. The paper is Mizokami : On characterizations of spaces with $G_\delta$-diagonals See its Theorem 1, also you can see the picture . ...