Let $X$ be a space; let $(Y,d)$ be a metric space. For the function space $Y^X$ one has the following inclusions of topologies: (Uniform) is finer than (compact convergence) is finer than (pointwise convergence).
Prove that if $X$ is compact the first two coincide and if $X$ is discrete the second two coincide.
I don't understand what I am supposed to prove. I mean isn't it obvious. If $X$ is compact, uniform convergence on compact set implies the uniform convergence on $X$. If $X$ is discrete, the only compact sets are the finite sets, and hence compact convergence is the same as pointwise convergence. Am I missing something?