1
$\begingroup$

I've been working through some reverse math with the completeness definitions of a metric space. More over, I've learned that in a metric space X that is ordered, The Least Upper Bound Property, Cauchy Criterion, Nested Interval Theorem, and Heine-Borel Theorem are all equivalent (provided that the Archimedean Property is true).

My Question: Let X be a metric space and suppose X is Cauchy complete. Does the Heine-Borel Theorem follow? In other words, is it true that if every Cauchy sequence converges to a limit in X, then every closed and bounded set in X is compact?

I've been able to show this is true for $\mathbb{R}^n$. Is this true for a general metric space?

$\endgroup$
4
$\begingroup$

No. Any infinite-dimensional Banach space is a counterexample to this. The closed unit ball in such a space is closed and bounded but not compact.

$\endgroup$
1
$\begingroup$

You already got a counterexample in another answer but let me add a potentially useful fact. If $X$ is a metric space then $Y\subseteq X$ is compact iff it is complete and totally bounded. In particular if $X$ is complete then $Y\subseteq X$ is compact iff it is closed and totally bounded. (Note that the closed unit ball in an infinite dimensional Banach space is bounded but not totally bounded)

$\endgroup$
0
$\begingroup$

A simple counter-example is the discrete metric, that is, $d(x,y)=1$ when $x\ne y,$ on an infinite set $X.$ Any $d$-Cauchy sequence is eventually constant and hence convergent. And every subset of $X$ is closed and bounded, but only the finite subsets of X are compact.

Another impediment is that if $d$ is a $complete$ metric on a set $X$ (that is, if every $d$-Cauchy sequence converges in $X$) then the metric $e(x,y)=\min (1,d(x,y))$ is also complete, and $e$ generates the same topology as $d$ does. Hence $d$ and $e$ generate the same collections of closed sets and of compact sets. So $X$ itself is closed and $e$-bounded, but not necessarily compact.

For example let $X=\Bbb R$ and $d(x,y)=|x-y|$ and $e(x,y)=\min (1,|x-y|).$

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.