I am working on a problem involving the rotation number of an orientation-preserving circle homeomorphism, which induces a map $\rho : \text{Homeo}^+(S^1) \to S^1$. It is well-known that this is a continuous map if we give $\text{Homeo}^+(S^1)$ the uniform topology.
Since $\text{Homeo}^+(S^1)$ (and more broadly, the full space of homeomorphism $\text{Homeo}(S^1)$) has a group structure via composition, I am wondering if it is a topological group, meaning that composition and inversion are coninuous maps.
More generally, if $(X,d)$ is a compact metric space, then $\text{Homeo}(X)$ is actually metrizable by way of the sup metric $\rho(f, g) = \max\{d(f(x), g(x)) : x \in X\}$ which induces the uniform topology. My Question: Is $\text{Homeo}(X)$ a topological group under composition and the uniform topology?
I think this is true, but I am having trouble showing it using $\epsilon-\delta$ with the sup metric. Does uniform continuity come into play at all?