Is the space of continuous maps $Top(X,Y)$ between two topological spaces compact if $X$ is?

Question

Suppose that $X$ and $Y$ are topological spaces, and we consider $Y^X$, i.e. the space of maps from $X$ to $Y$ with the compact-open topology. If $X$ is compact then can we say anything about $Y^X$?

Thanks! Jon

Answer 1

Suppose $X$ is a one-point space. Then $Y^X$ is naturally homeomorphic to $Y$, so is not compact unless $Y$ is.