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**.

- Given any sequence of points (x
- 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
topological spaces make it a powerful abstraction mechanism.*all*