# Locally compact Hausdorff space is metrizable

Given $X$ a Hausdorff space, I have a hunch that

$X$ is locally compact $\iff X$ is metrizable.

I am not sure if it is true because I do not know how to prove that. To prove the implication (locally compact Hausdorff space is metrizable) I guess I need either $X$ is second countable or locally metrizable.

Could some one prove it or otherwise, give a counter example that the statement is not true. Thanks!

• I don't think the equivalence is true, and you probably can find some answers here math.stackexchange.com/questions/74923/… Commented Apr 11, 2016 at 14:05
• Counterexample: $[0,1]^S$ for uncountable $S$. That's a compact Hausdorff space that's not metrizable. Commented Apr 11, 2016 at 14:10
• @Wore&DavidCUllrich,, yup. It's false statement. Thanks for the example. Commented Apr 12, 2016 at 9:38

Both @Wore's and @David C Ullrich's comments above (and the linked question) give you examples of locally compact $T_2$ spaces which are not metrizable.
We consider the real line $\mathbb{R}$ with the Euclidean topology, and the irrationals $\mathbb{I}$ as a subspace with the inherited topology. The irrationals are metrizable (by the Euclidean metric) but not locally compact (can you prove this?).
• If $\mathbb{I}$ were locally compact, then for every irrational $r\in\mathbb{I}$ has compact neighbourhood, i.e. we can find a nbhd $U_r$ of $r$ where every sequence has a convergent subsequence. But since rationals are dense in $\mathbb{R}$ there always exists a rational $x \in U_r$, then the sequence approximating $x$ won't have a convergent subsequence. For e.g., the interval contains 1/2, since both rationals are not order-complete, there cannot be a sequence of irrationals converging to 1/2. So, the irrationals are not locally compact. Commented Apr 12, 2016 at 2:32
• Thanks for the refference. That book seems very useful. I will get that book. Thanks again! So, my hunch is not true then. Now I will raise another question, haha. If $X$ is Haussdorf, second countable, and metrizable, is it always locally compact then? Commented Apr 12, 2016 at 2:38