TITLE: Can we get un-obvious results by defining sophisticated topologies?
QUESTION [15 upvotes]: What I originally found so interesting about general topology was how general a type of thing a topology is, and how the terminology open, closed, compact, continuous, convergence et cetera means something completely different depending on which topology we are using. It seemed to me that each result was actually one result PER choice of topology. But the course I did mostly focused on real space and the Euclidean topology, the boundedness principle and equivalence of norms. I admit all of these are practically useful, but they were not what made topology seem so interesting.
My question is this: Are there any cases where we engineer an unusual topology with an eye to proving specific unobvious results. The work would then be in proving it is a topology, exactly what continuous functions look like, what compactness is and so on. And then invoke a general theorem to get the result?

REPLY [3 votes]: André Weil imagined what would happen if a nonsingular projective algebraic variety over a finite field had a topology which satisfied certain properties (i.e. had a "good" cohomology theory, whatever that is). Using such an assumption, he was able to prove extremely important results about counting the number of points on such varieties (over a finite field, the number of points is always finite). These were first conjectured in the special case of curves by Artin. The problem was that such a topology with a good cohomology theory was not known. The Weil conjectures were born.
Weil was able to prove the conjectures for curves but there was really no hope for general varieties of dimension $> 1$. Grothendieck defined a new topology largely for the goal of proving the Weil conjectures. This is not exactly a topology, it is what is known as a Grothendieck topology. The gist of it is that the most important thing (in algebraic geometry) is that a topology allows you to speak of covers and refinements of covers. By only requiring a "topology" to satisfy axioms relating to covers, one is allowed more flexibility. With this flexibility in mind, Grothendieck defined and developed several topologies and cohomologies, most notably the étale topology and I-adic topology.
By mostly proving theorems for these topologies which parallel well known theorems for more "usual" topologies, a twenty or so year program was able to prove most of the Weil conjectures. Only the Riemann hypothesis remained. Grothendieck outlined a program of standard conjectures which would ultimately lead to a proof of the Riemann hypothesis. These remain largely open. Deligne was able to prove the Riemann hypothesis using a completely different argument which did not parallel well-known theorems in "usual" topology.  He was awarded the fields medal for his proof.