Rudin, in Principles of Mathematical Analysis, defines compactness: A set $E$ in a metric space $X$ is compact if and only if for any open cover $\{G_\alpha\}$ of $E$ there exists a finite sub-cover $G_{\alpha_1},...,G_{\alpha_k}$ such that $$E \subseteq G_{\alpha_1} \cup \cdots \cup G_{\alpha_k}.$$
My questions is why must the the cover be open sets and not just sets in general? What are some important theorems of compact sets that depend on the open covers being open?
If we can cover $X$ by any subsets we could use the cover $\{x\}: x \in X$ and we could only cover by a finite sub-cover iff the covered set is finite.
So the only compact subsets would be finite. In a metric space (and in many topological spaces generally) all singleton sets are closed, so even restricting to closed covers would have the same effect.
It's not an arbitrary choice: it was discovered that certain subsets of $\Bbb R^n$ behaved nicer than others (continuous functions are bounded on it, every sequence in it has a convergent subsequence) and then it was found that the essential property that made all these proofs of those properties work, was this open cover definition which turned out to be equivalent (for metric spaces) to these properties. This open cover formulation has the added advantage that it behaves very well in general spaces too, not just metric ones.