0
$\begingroup$

In "Understanding Analysis", Abbott defines a compact set as a set $K \subseteq \mathbb{R}$ in which every sequence has a subsequence that converges to a limit which is also in $K$. Shortly after, he introduces the Heine-Borel theorem which states that a set is compact iff it is closed and bounded.

To me it seems that the latter condition is much more natural as a definition. Although I understand the sequential condition as valuable in proofs of other facts (it is useful in the proof of preservation of compact sets for continuous functions), why is it used as the definition? The properties of closedness and boundedness and are more obvious in the simplest example of a compact set $[a,b]$, so what are some weird compact sets where the sequential definition is more obvious/useful?

$\endgroup$
  • $\begingroup$ In some cases, being closed and bounded is the definition of compact. Usually one definition fits better than the other in the context it's used. $\endgroup$ – Bill Wallis Sep 4 '18 at 0:52
1
$\begingroup$

The main reason why "compactness" is not defined this way is that it has meaning in more general spaces than $\mathbb{R}$. The concepts of "sequences", "subsequences", and "convergence" can be talked about in more general settings, as indeed can the terms "closed" and "bounded", but as we generalise, we find that closed and bounded sets are not always compact (though the converse happens to be true).

When we do look at generalisations, the definition with sequences and subsequences turns out to be the more natural definition. It's the definition you use, for example, when proving fundamental real analysis theorems like the extreme value theorem. Proving it directly just from closedness and boundedness is messy, tricky, and uses copious amounts of the supremum axiom, whereas using compactness is elegant. And, indeed, the extreme value theorem works in these more general spaces too, provided you define compactness in terms of sequences.

If you want to know more about these more general spaces, you can look up "metric spaces". Basically, a metric space is a set with a function that takes two points in the set, and spits out a positive number that could be interpreted as a distance between these points. From this, there are natural definitions of convergence, closedness, and boundedness, etc.

I should also point out that there is a further generalisation, called topological spaces, but these are far more abstract. In this setting, there is actually another, different definition for compactness, which turns out to be more useful (but still, the extreme value theorem holds!).

$\endgroup$
1
$\begingroup$

The difference in definition comes down to how general you want to be while still maintaining the key properties of the definition. I personally believe that the best definition is the topological one, which states that a space X is compact if given any open cover of X there is a finite subcover of it. Why exactly I prefer this is imprecise, but I have found this to be the best balance of generality and strength. The definition by subsequences (which I will refer to as sequential compactness) is more general than the definition as a closed and bounded set. The reason is that the latter is equivalent to compactness for subsets of $ \mathbb{R}^n $ but the former is equivalent for a general metric space. Of course, it's good to keep all equivalent forms in mind when working with compactness, or anything for that matter.

$\endgroup$

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.