TITLE: Topological groups, why need them?
QUESTION [15 upvotes]: I'm reading through Munkres and Armstrong's books on topology. However, I find topological groups to be really complicated objects!
I feel they are twice as hard to deal with then just groups and topological spaces, but why do we need it? 
  Can anybody help me?

REPLY [22 votes]: To a topologist, topological groups are interesting in their own right. The group structure actually gives us interesting topological structure, too! One interesting fact is that the fundamental group (an important topological invariant) of a topological group is Abelian, a fact that spectacularly fails to be true in general - any group can be the fundamental group of some space (proof).
To an analyst or a number theorist, certain kinds of topological groups have incredibly useful properties. Locally compact groups are known to have a Haar measure, a measure that (in some sense) respects the group operation (proof; I can't advise reading it, honestly - it's not one of my favorite proofs). A group having a Haar measure means we can import topics from measure theory and essentially do analysis on this group (in particular, Fourier analysis). One important application of this is in algebraic number theory, where the Haar measure lets us do a lot of things we wouldn't be able to do otherwise.
I've heard that the representation theory of topological groups is particularly nice, but I don't have a reference for this that I've read, so take it as you will; but this fact might make them interesting to an algebraist. In particular, one can recover quite a lot of the representation theory of finite groups by working instead with compact groups; there's a powerful proof technique where one 'averages' over the group, and the Haar measure lets one do this for compact groups, too.
The combination of the group structure and the topological structure is very powerful, and makes these objects helpful to mathematicians of all stripes whenever they appear!