One of the most useful examples of pointless topologies is the étale topology (the major work on this was done by Alexander Grothendieck and Mike Artin). Before I say more it is very important that you know that in French the word étale refers to the appearance of the sea in certain types of weather. It's all becoming much clearer isn't it?

OK why do we need this "topology". Start with an algebraic variety i.e. the zero locus of some polynomials in kn, for a field k (or projective space). These varieties have a natural topology, the closed sets for the topology are given by subsets which are also defined by the vanishing of polynomials. This topology doesn't have a lot of open sets though, it is pretty coarse. If the field we work over is C then there is a competing topology (just the usual one for subsets of Cn or projective space) called the analytic topology which is much finer.

However, if we are studying algebraic number theory then the field is not C, it might be, for example, (the algebraic closure) of Zp (the ring of integers modulo p).

The étale topology gives us a way of faking the finer analytic toplogy in these more general situations. Basically it works like this. As you can imagine just as rings have ring homomorphisms and groups have group homomorphisms there are morphisms between algebraic varieties. The open sets for the étale topology are étale morphisms. (If the field is C then a morphism is étale iff it is a local isomorphism in the analytic topology.) It's not too hard to check that the axioms for a pointless topology are verified. Other examples include the faithfully flat topology.

If you got this far congratulations, you are at the forefront of mathematics. If your head hurts, don't blame me ariels made me do it!