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
Best Answer
The argument for 2 is correct.
For 1, you can show that $Y$ has a countable base as well: as $X$ is locally compact and second countable, it has a countable open base $\mathcal{B}$ such that $\overline{O}$ is compact for all $O \in \mathcal{B}$.
Then the point at infinity $\infty$ has a local base of the form $\{\infty\} \cup \{X \setminus C: C = \cup_{i=1}^n \overline{O_i}, O_i \in \mathcal{B}\}$, which is countable (little argument requireed). Show it is a local base for $\infty$ (which uses that $X$ is lcoally compact), and combine it with $\mathcal{B}$ to form a countable base for the whole compact Hausdorff $Y$, which is then metrisable by Urysohn again.