TITLE: Please list a few topological groups that I should learn about.
QUESTION [6 upvotes]: 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.

REPLY [3 votes]: 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).