A space $X$ is locally metrizable if each point $x$ of $X$ has a neighborhood that is metrizable in the subspace topology. Show that a compact Hausdorff space $X$ is metrizable if it is locally metrizable.
Hint: Show that $X$ is a finite union of open subspaces, each of which has a countable basis.
I tried to use the fact of compact space. But I do not know if the opens are compact subspaces.