Is the space of continuous functions compactly generated when the space is?

Question

Let $X$ be compactly generated, i.e. a subset $A$ of $X$ is open in $X$ iff $A\cap C$ is open in $C$ for any compact subspace of $X$; and let $(Y,d)$ be a complete metric space. Is it true that the space $\mathcal C(X,Y)$ is compactly generated?

Here $\mathcal{C}(X,Y)$ denotes the space of continuous functions $f:X\to Y$ with the supremum metric $\overline\rho(f,g)=\sup \{ \overline d(f(x),g(x)):x\in X\}$, where $\overline d$ denotes the standard bounded metric of $(Y,d)$.

Answer 1

$\mathcal{C}(X,Y)$ is by your definition a metrisable space. Any first countable space is compactly generated. So the answer is yes, regardless of what $X$ is.

It might be more interesting to consider $C_p(X)$ (so the real-valued functions on $X$ in the pointwise (aka product) topology); IIRC that is a classic question. Or maybe the compact-open topology might be more relevant in this context.