compact (idea)

What isn't clear from the above is why anyone bothered to define the notion of compactness: in my opinion it is an idea which perfectly embodies the notion of mathematical elegance.
I suppose a deep reason why the notion of compactness is useful comes from the defining axioms of topological spaces: that any finite intersection of open sets is open, but that arbitary intersections are not (eg. consider the intersection of all the open sets (-ε,ε) for all ε between 0 and 1 - the intersection is {0}, which is not open). Given compactness, we can reduce the arbitary intersections to finite ones and then there is no problem.

Here are a few nice consequences of compactness:

• Proofs get a lot shorter. Compare these two proofs that a continuous image of [ 0,1 ] is closed and bounded:
  • Given any sequence of points (x_n) in [ 0,1 ] , it will have a subsequence converging to some point in [ 0,1 ] (because closed and bounded). Knowing this, one can show that the image set also has convergent subsequences to arbitary sequences too: let (y_n) be such a sequence in the image set, and then for each y_n define a x_n such that f(x_n)=y_n. The given the convergent subsequence of the (x_n), can define one for the (y_n), which agree in the limit by the continuity of f.
    The subsequence property on the image set, one can show that it is closed and bounded: if not could produce a sequence (y_n) such that |y_n| > n for every n, which could not have a convergent subsequence. Nasty.
  • [ 0,1 ] is closed and bounded, so compact. The continuous image of a compact set is compact - and hence too closed and bounded. Nice.
  In this way, you can prove many similar things quickly by simply stringing together known lemmas about compact spaces.
• You can make lots of spaces compact simply by adding an extra point (and making a change to the topology)- eg. adding 'infinity' to the complex plane turns it into the Riemann Sphere. Doing this gives you a sensible way to deal with infinity, and gives rise to the notion of Riemann Surfaces, which are things of great beauty.
• Its simple formulation and applicability to all topological spaces make it a powerful abstraction mechanism.