That is awesome, could you maybe just give a hint about what kind of area it was in? I took topology last year and it was fun but I could never imagine it applied to CS.
Now generalise into a mapping from C^2 -> [0,1] and think about continuity. Various properties of objects moving in your images translate nicely into properties of the functions. The thing we were trying to do turned out not to be posible, but this way of thinking showed us a similar idea that worked, and the solutions provided could be warped into a solution of the original problem.
Interestingly, in a sense it was like proving that a given problem is NP-Complete in that we converted the problem we had into another, then converted the solutions back.
It's the rich inter-connectedness that gives you this, not the individual fields. People recommend that you learn Linear Algebra, or Complex Analysis, or Group Theory, or Topology, but in truth, no single field helps on its own.
They are intertwingled, and it's the connections that carry the real benefits.