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Let $\delta$ be the discrete metric on a non-empty set $X$. Characterize the subsets of $X$ which are compact in $(X, \delta)$.

I remember the answer from a previous class: $A \subseteq X$ is compact $\iff A$ is finite.

I've proven one way in the general:

Suppose $S$ is a finite set. Since there are only a finite number of terms, no sequence of distinct terms exists strictly in $S$, so $S$ does not have any limit points (and thus contains all of them; i.e. $\emptyset$). Furthermore, since $S$ is finite, there exists a well-defined maximum $M = \max \{|s| : s \in S\}$. Thus, $S$ is bounded by $M$. Since $S$ is closed and bounded, it is compact.

but I can't seem to go the other way. How can I prove that, if $A \subset (X, \delta)$ is finite, then it is compact?

I had a look at Show that in a discrete metric space, every subset is both open and closed., but note that an infinite set can most certainly be bounded.

SpiderCrawler
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Sean Allred
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  • You probably meant to ask "how can I prove that if $A$ is compact in $X$, then $A$ is finite. – Ittay Weiss Apr 28 '13 at 12:36
  • @IttayWeiss Good point. – Sean Allred Apr 28 '13 at 12:36
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    Do you know what is the definition of a space being compact? being closed and bounded is not the definition, is wrong in general. – Dennis Gulko Apr 28 '13 at 12:41
  • @DennisGulko You're right, I think the closed and bounded thing only applied in $\Re^n$ - but I don't really know how to work with the 'proper' definition of compactness: every sequences has a convergent subsequence in the set. – Sean Allred Apr 28 '13 at 12:45

5 Answers5

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First, in a metric space, a subset is compact if, and only if, every sequence of distinct elements in it has a convergent subsequence with limit in the subset. So, your argument for the first half can be slightly simplified (and in fact corrected, since the criterion you use for compactness is not quite right):there are no sequences of distinct elements in a finite subset, thus a finite subset is compact.

Now, assume $A$ is compact in $X$. Suppose $A$ were infinite. Then there is some infinite sequence of distinct elements of $A$. By compactness, this sequence has a convergent subsequence with limit $L\in A$. But in the district metric, the only convergent sequences are the eventually constant ones. No subsequence of a set of distinct elements is eventually constant, thus a contradiction.

Remark: the topological formulation of compactness (which is equivalent to the metric one (for metric spaces that is)), that a set is compact if, and only if, every open cover of it has a finite subcover gives the same result a lot quicker. If $A$ is compact, then cover it by its singletons. In the discrete metric, every singleton is open, so this is an open cover. By compactness, $A$ must be covered by a finite number of its singletons, so is itself finite. Conversely, if $A$ is finite and an open cover is given, then dilute the open cover to contain, for each element of $A$, just one open set containing that element. This gives a finite subcover.

Ittay Weiss
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  • That's the missing piece! I knew the eventually constant thing, but forgot that compact sets must have convergent subsequences of distinct elements! – Sean Allred Apr 28 '13 at 12:47
  • (I wrote the proof in my answer several months ago in another class, and didn't really read it over again.) – Sean Allred Apr 28 '13 at 12:56
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Hint Every $\{a\}$ is open hence $$A\subset\cup_{a\in A}\{a\}$$ is an open cover of $A$ and then there's finite subcover...

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In a metric space, it is false in general that a bounded closed set is compact (for a counterexample, consider $\{q \mid 2< q^2<3\}$ in $\mathbb{Q}$).

You can prove that a finite set is always compact in a metric space using open coverings or subsequences. For the converse in discrete metric space, you can exhibit a sequence without converging subsequence in every infinite set (or exhibit an open covering whitout finite subcovering).

Seirios
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  • You're right, this only applies to the reals (Or some complete set, or some other extra quantifier that I tend to take for granted.) – Sean Allred Apr 28 '13 at 12:55
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Given any open cover of of any finite set, can be easily found a finite sub cover of the set. Then $A$ finite implies $A$ compact.

Now, in the discret metric no set have limit points. Then $A$ compact and $A$ infinite can't be hold, for if $E$ is an infinite subset of $A$ then $E$ must have a limit point in $A$.

DIEGO R.
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Hint: Show that a sequence in $(X, \delta)$ converges if and only if it is eventually constant.

Michael Albanese
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