I need to show that: If $X$ is locally compact, second countable and Hausdorff, then $X^*$ is metrizable and hence $X$ is metrizable.
I have already showed that every locally compact Hausdorff space is regular. Then I use that $X$ is regular and second countable to apply Urysohn's metrization theorem and say that $X$ must be metrizable. However, this doesn't show that $X^*$ (the one point compactification) is metrizable. I know that $X^*$ is compact and regular, but I don't know that $X^*$ is second countable. How can I show that $X^*$ is second countable and hence metrizable? Or, if it is not possible, how else should I approach this problem?