Let $X$ be a locally compact Hausdorff space.Let $Y$ be the one-point compactification of $X$. Two questions are:
- Is it true that if $X$ has a countable basis then $Y$ is metrizable?
- Is it true that if $Y$ is metrizable then $X$ has a countable basis?
My attempt:We know that every compact space which is metrizable has a countable basis.Thus in (2) we have $Y$ is $2^{nd}$ countable and a subspace of a $2^{nd}$ countable space being $2^{nd}$ countable so $X$ is $2^{nd}$ countable .
In (1) I could only figure out that $X$ is regular since it is locally compact Hausdorff space.Also $X$ has a countable basis so by Urysohn Metrization Theorem $X$ is metrizable.
But how can this help me conclude whether $Y$ is metrizable/not?
Any help will be helpful