# Pathological examples of compact sets in analysis

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?

• 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. – Bill Wallis Sep 4 '18 at 0:52

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).
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.