I've noticed that it's often interesting to consider what types of properties of a topology are preserved or lost when mapping between different types of spaces. This led me to wonder about spaces of functions themselves.
In this case, suppose I have two metric spaces, $(S,c)$ compact and $(T,d)$ complete, and I denote by $C(S,T)$ the set of all continuous functions from $S\to T$. I can put a metric on $C(S,T)$ defined by $\rho(f,g)=\sup_{s\in S}d(f(s),g(s))$. With this in place, is it true that $(C(S,T),\rho)$ is also complete?