Now I would like to change question in If $X$ is locally compact, second countable and Hausdorff, then $X^*$ is metrizable and hence $X$ is metrizable.
Is it true that
If $X$ is Hausdorff, second countable, and metrizable, then $X$ is locally compact.
To show that $X$ is locally compact, need to show that for every $x\in X$ has compact neighbourhood, i.e. we can find a nbhd $U_x$ where every sequence has a convergent subsequence. I raise this question because my hunch at Locally compact Hausdorff space is metrizable is not true.