We know that Heine-Borel Theorem does not hold in arbitrary metric space. (If $A$ be an infinite subset of a discrete metric space then $A$ is not compact though it is closed and bounded.) How do we characterise the class of metric spaces that satisfies the Heine-Borel Theorem?

I have read the proof of Heine-Borel Theorem in the Real line. It considers any closed and bounded subset $[a,b]$. Then it attains an open cover $U$ (say). Let $P=\{x\in[a,b]:U$ has a finite subcover for $[a,x]\}$. We take it $V$. Clearly $P$ is bounded above and hence it attains supremum...say $c$. Let $c$ contain in $U_1$ (element of $U$). We take an open ball centered at $c$ that is contained in $U_1$. Then the open ball centered in $c$ with the union of $V$ is also finite open sets. Then $c$ coincides with $b$ and hence $[a,b]$ becomes compact.

Which step does not hold in arbitrary metric space? (I have written the procedure of the proof briefly.)

  • $\begingroup$ $P$ already makes no sense in a general metric space, even $\mathbb{R}^n$ (where Heine-Borel still holds). You might try imitating the proof of Heine-Borel in $\mathbb{R}^n$ instead to see where that fails. $\endgroup$ – Ian May 23 '17 at 20:10
  • $\begingroup$ In most metric spaces there is not a given total order the way there is in $\Bbb{R}$. The step of taking a supremum thus fails. $\endgroup$ – ForgotALot May 23 '17 at 20:11
  • $\begingroup$ #ForgotALot Do you mean that any metric spaces except ordered sets don't hold this property? $\endgroup$ – Subhajit Saha May 23 '17 at 20:42
  • $\begingroup$ @SubhajitSaha It doesn't mean that the property fails, but this proof makes no sense outside of an ordered set, because there's no analogue of $P$. $\endgroup$ – Ian May 23 '17 at 20:54
  • $\begingroup$ en.wikipedia.org/wiki/Heine%E2%80%93Borel_theorem $\endgroup$ – Mirko May 24 '17 at 1:39

You need a complete metric space, with the topology induced by the metric, so that a subset of the metric space can be closed, i.e. contain all cluster points.

Also for metric space, the concept of being bounded becomes complicated. In higher dimensions of ℝ^(𝑛) you lose well-ordering (is [-5, 6] greater than or less than [-6, 5]?) so you need another concept of boundedness.

Also you could define a bounded set that is infinite, so for generalized metric space you should use or build up to the property of total boundedness, that a finite number of balls can cover the set. This is very close to being compact, and then you can go from there (show it is closed).


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