What is the Purpose of Topology in Mathematics?

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I have an idea of what topology is but I am clueless as to what applications it has? Anybody have any idea what topology is used for?
 
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It used to have little, but now physics is discovering more and more things that depend on topology (mostly homotopy and connectedness). The Aharonov-Bohm effect and Berry's phase, two much studied experimental effects, depend on the non simple connectivity of the configuration space.

And topology of fibrations is being much applied in modern theoretical physics.
 
Topology pays a major role in string theory.
 
Not an "application" in the sense of an application to science, but the purpose of "topology" is to generalize the idea of "continuous". The most general mathematical object in which one has a notion of "limit" and "continuous function" is the topological space.
 

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