So I'm trying to find the most general setting in which I can talk about completeness of function spaces. In metric spaces, it's simple to show that the space $C(X,Y)$ of continuous functions $X\longrightarrow Y$ is complete whenever $X$ is compact and $Y$ is complete.
I have a hunch that $C(X,Y)$ is complete if $X$ is compact and $Y,\mathcal U$ is a complete uniform space. First of all, we can generate a uniform structure $\mathcal F$ on $C(X,Y)$ by declaring that for each entourage $U\in\mathcal U$, there is an entourage $F\in\mathcal F$ defined by $$(f,g)\in F \iff \forall x\in X, (f(x),g(x))\in U.$$ (Note: I am open to the fact that another formulation of uniform spaces might be more appropriate here, particularly the pseudometric formulation. Feel free to demonstrate this!)
My questions are: (i) under what conditions is $C(X,Y)$ complete, and (ii) does the uniform topology on $C(X,Y)$ coincide with the compact-open topology?