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!