Please list a few topological groups that I should learn about. I'm going through Munkres' Topology book and there's a lot about topological groups.   For fear that I'll forget the theorems on them I'd like to connect each thing I prove with a real-world example.  Please list some topological groups or rings that are interesting besides the obvious ones like $(\mathbb{C}, +, \cdot)$, and give a reason why they're interesting.  Thanks.
 A: *

*All of the classical Lie groups. This one should be self-explantatory. This is probably closest to the type of answer you were thinking about.

*Local fields and their valuation rings (this might be a little advanced). They are a huge part of modern number theory.

*In general, the matrix ring or the general linear group over a topological ring. This obviously shows up everywhere, from calculus to number theory.

*Definitely the circle group $\mathbb{T}=\{z\in\mathbb{C}:|z|=1\}$. This is a topological group that comes up in multiple places in analysis, in particular it's huge in representation theory. It's also important to take a glance at general tori groups $\mathbb{T}^n$.

*Lattices in real vector spaces. They have trivial topological structure, but seeing how to prove this/exploit this will give you a good sense of how topological group arguments work.

*Inner product spaces, either finite dimensional or Hilbert spaces. They are a foundational part of modern analysis.

*As Zhen Lin points out, we should add in profinite groups (inverse limits of finite groups). In particular Galois groups of (for interests sake, let's say infinite) extensions.

A: The adèles and idèles of a global field are both fundamental to modern number theory. With these topological groups (and their associated Haar measures) it becomes possible to give a direct, unified proof of the finiteness of class number and the Dirichlet unit theorem, though this is just the beginning of their utility. I also suggest looking at the MathOverflow threads What problem do the adeles solve? and Who fixed the topology on ideles?
A: Also: for a topological space $X$, the group $\mathrm{Homeo}(X)$ (with compact-open topology) and, for a smooth manifold $M$, $\mathrm{Diff}(M)$, with topology of uniform (on compacts) convergence of all partial derivatives with respect to a fixed background Riemannian metric on $M$. Other similar and important groups are the groups of symplectomorphisms and Hamiltonian symplectomorphisms (of symplectic manifolds), contactomorphisms (of contact manifolds) and biholomorphic automorphisms (of complex manifolds).  
