Why is compactness so important? I've read many times that 'compactness' is such an extremely important and useful concept, though it's still not very apparent why. The only theorems I've seen concerning it are the Heine-Borel theorem, and a proof continuous functions on R from closed subintervals of R are bounded. It seems like such a strange thing to define; why would the fact every open cover admits a finite refinement be so useful? Especially as stating "for every" open cover makes compactness a concept that must be very difficult thing to prove in general - what makes it worth the effort?
If it helps answering, I am about to enter my third year of my undergraduate degree, and came to wonder this upon preliminary reading of introductory topology, where I first found the definition of compactness.

Compactness does for continuous functions what finiteness does for functions in general.
If a set $A$ is finite, then every function $f:A\to \mathbb R$ has a max and a min, and every function $f:A\to\mathbb R^n$ is bounded.  If $A$ is compact, then every continuous function from $A$ to $\mathbb R$ has a max and a min and every continuous function from $A$ to $\mathbb R^n$ is bounded.
If $A$ is finite, then every sequence of members of $A$ has a sub-sequence that is eventually constant, and "eventually constant" is the only sort of convergence you can talk about without talking about a topology on the set.  If $A$ is compact, then every sequence of members of $A$ has a convergent subsequence.
