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

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 $$(Y,d)$$.

• What have you tried? Aug 12 '20 at 19:18

$$\mathcal{C}(X,Y)$$ 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.