Let $x\in X$ . Let $U$ be an open set containing $x$ such that it is metrizable in the subspace topology. Then $U$ is locally compact (Corollary 29.3) and Hausdorff. Let $V$ be a neighborhood of $x$ in $U$ such that $\overline{V}_U$ is compact. Then $\overline{V}_U=\overline{V}\cap U$ is a compact Hausdorff metrizable subspace containing $x$ . Using Exercise 3, we conclude that the subspace topology of $\overline{V}\cap U$ is generated by a countable basis. $V\subseteq\overline{V}\cap U$ is open in $X$ and second-countable as well (in the subspace topology). Now cover $X$ with such neighborhoods for all points and find a finite subcovering $V_n$ . Consider the finite union $\mathcal{B}$ of all countable subspace bases. Suppose $y\in U$ . Then there is $V_n$ containing $y$ . $V_n\cap U$ is a neighborhood of $y$ in $V_n$ , and it contains a basis sub-neighborhood that is open in $X$ and belongs to $\mathcal{B}$ . So, $X$ is second-countable, therefore, according to Exercise 3, it is metrizable.