Let $X$ be a compact metric space and let $Y$ be a non-metrizable topological space. How can we show that $C(X,Y)$ with compact-open topology is non-metrizable?
I was thinking of embedding $Y$ into $C(X,Y)$ via $y\mapsto E_y$ where $E_y$ is the evaluation map $f\mapsto f(y)$. Then arguing by contradiction. However, that's as far as I've gotten.
Instead of the evaluation map, which maps $C(X,Y)$ to $Y$, use the map $c$ which maps $y \in Y$ to the constant map $c(y):X\to Y$, with value $y$. This is an embedding of $Y$ into $C(X, Y)$. It’s well-defined as constant maps are continuous and $c$ is clearly 1-1 as well. Just check continuity of $c$ and its openness as a map from $Y$ onto its image. ($c^{-1}[[K,O]] = O$ and $c[O] = [X,O] \cap c[Y]$ e.g.)