Let $\mathscr{C}$ be the space of conformal transformations of $\mathbb{R}^n$, and say I have some functional $F:\mathscr{C}\to\mathbb{R}$. (By conformal transformations, I mean angle-preserving diffeomorphisms of $\mathbb{R}^n$.)
I want to show that $F$ is continuous. In order to do this, I need to know something about the topology of $\mathscr{C}$.
That said, is there any standard topology or metric on $\mathscr{C}$? If not, what would be a sensible way to topologize this space?
There is a general theorem that for $n \ge 2$ the conformal diffeomorphisms of $\mathbb{R}^n$ are exactly the similarity transformations, generated by reflections and homotheties. In the case of $n=2$ this is a theorem in any first course in complex analysis. In the case of $n \ge 3$ this is known as Liouville's Theorem on conformal mappings which is actually much stronger, namely, any conformal diffeomorphism between two connected, open subsets of $\mathbb{R}^n$ is the restriction of a similarity transformation.
From this, it follows there are a few equivalent descriptions of a very natural topoogy on the group of conformal diffeomorphisms. One simple description of this is the compact open topology. But you can also use the fact that this group has a faithful matrix representation in $GL(n+1,\mathbb{R})$. Indeed, the whole group of affine transformations of $\mathbb{R}^n$ has a faithful matrix representation into $GL(n+1,\mathbb{R})$. Then you can simply use matrix coordinates to define the topology, i.e. use the embedding into $\mathbb{R}^{(n+1)^2}$.