Is every metrizable topology induced by a Heine-Borel metric?

Question

A metric space $(X,d)$ has the Heine-Borel property if for any subset $A$ of $X$, $A$ is compact if and only it is closed and bounded. ($\mathbb{R}^n$ is the classic example.) My question is, for any metrizable topological space $X$, does there exist a metric on $X$ which induces the topology on $X$ and which has the Heine-Borel property?

If not, what is an example of a topological space such that all the metrics which induce the topology fail to have the Heine-Borel property?

score 3 · Accepted Answer · answered Dec 19 '18 at 06:13

3

Any metric space with Heine Borel property is sigma compact. An infinite dimensional Banach space is never sigma compact.

answered Dec 19 '18 at 06:13

Kavi Rama Murthy

311,013

Or any uncountable set with the discrete topology, for the same reason. – A. Pongrácz Dec 19 '18 at 06:16
Why does the Heine-Borel property imply sigma-compactness? – Keshav Srinivasan Dec 19 '18 at 06:19
$X=\cup_{n=1}^{\infty} \overline {B} (x,n)$ for any fixed point $x$. @KeshavSrinivasan – Kavi Rama Murthy Dec 19 '18 at 06:21
OK, that makes sense. By the way, does every sigma compact metrizable space have a Heine-Borel metric which induces its topology? – Keshav Srinivasan Dec 19 '18 at 06:29
It must also be locally compact. $\mathbb Q$ is $\sigma$-compact but not locally compact. Maybe a locally compact separable metrizable space has a Heine–Borel metric? – bof Dec 19 '18 at 09:17
@bof Does having a Heine-Borel metric imply seperability? – Keshav Srinivasan Dec 19 '18 at 19:16
@KeshavSrinivasan For metric spaces, Heine–Borel $\implies$ $\sigma$-compact $\implies$ Lindelof $\iff$ separable. – bof Dec 19 '18 at 22:05
In my opinion boundedness and Heine -Borel property are poorly defined concepts in a general metric space. This is because any metric $d$ is equivalent to $\frac d {1+d}$ which makes the entire space bounded. @KeshavSrinivasan – Kavi Rama Murthy Dec 19 '18 at 23:07
@bof I just posted a question about your guess: https://math.stackexchange.com/q/3047238/71829 – Keshav Srinivasan Dec 20 '18 at 06:42

Is every metrizable topology induced by a Heine-Borel metric?

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