A Universe of Sorts

If $X$ is a compact space and $Y$ is a metric space, consider two functions $f, g: X \to Y$ .
We can define a distance $d(f, g) \equiv \min_{x \in X} d(f(x), g(x))$ .
The $\min_{x \in X}$ has a maximum because $X$ is compact.
Thus this is a real metric on the function space $Map(X, Y)$ .
Now suppose $Y$ is no longer a metric space, but is Haussdorf. Can we still define a topology on $Map(X, Y)$ ?
Let $K \subseteq X$ be compact, and let $U \subseteq Y$ be open such that $f(K) \subseteq U$ .
Since $Y$ is Hausdorff, $K \subseteq X$

§ Motivation for the Compact-open Topology