# Locally compact 2nd countable Hausdorff space and complete metrizability

I was recently trying to verify certain things in the setting of Locally compact 2nd countable Hausdorff spaces. I thought that this is a natural collection of spaces more general than metric spaces, but I think that following the metrization theorems that such spaces must be metrizable with a complete metric. I didn't find such an explicit statement, so I thought to check whether this is indeed correct.

My reasoning\intuition is as follows.

• A locally compact Hausdorff space is completely regular.
• By Uryshonn's metrization theorem, a completely regular 2nd countable space is metrizable.
• Every 2nd countable locally compact Hausdorff space is also $$\sigma$$-compact.
• For such a space, closure of balls should be compact.

The last point is imprecise, but I have the strong suspicion that such a statement must hold. I assumed if I am wrong someone can generate a well known counter-example. Also if this statement is correct, this should be known under some terminology which I don't know how to look for.

I would appreciate any directions on this.

It's certainly not true that in a second-countable locally compact metric space, closures of balls are compact. For instance, consider $$(0,1)$$ with the usual metric.
Instead, you have to find a way to construct a special metric for which this is true. Here is one way to do so. Let your space be $$X$$ and let $$X^*=X\cup\{\infty\}$$ be its $$1$$-point compactification. If $$(U_n)$$ is a countable basis for $$X$$ consisting of sets with compact closures, then there is a countable local base at $$\infty$$ in $$X^*$$ consisting of the complements of the closures of finite unions of the $$U_n$$'s (since a neighborhood of $$\infty$$ is the complement of a compact set in $$X$$ which must be covered by finitely many of the $$U_n$$'s). Combining this countable local base with $$(U_n)$$, you get a countable basis for $$X^*$$, so $$X^*$$ is also metrizable. Now take a continuous function $$f:X^*\to\mathbb{R}$$ that vanishes only at $$\infty$$ (e.g., $$f(x)=d(x,\infty)$$ for some metric $$d$$) and embed $$X$$ into $$X\times\mathbb{R}$$ by $$x\mapsto(x,1/f(x))$$. This gives a metric on $$X$$ with the property that any bounded set has compact closure (since a set where $$1/f$$ is bounded is contained in the complement of a neighborhood of $$\infty$$ in $$X^*$$).
This pi-Base search confirms that all Weakly Locally Compact + Second Countable + $$T_2$$ spaces must be Completely Metrizable (by way of chaining together 10 different theorems).
A bit more directly: $$T_2$$ + weakly locally compact gives regularity, and with second countability we have metrizability. Finally, all weakly locally compact and metrizable spaces are completely metrizable.