# What is a mild sufficient condition on $X$ such that $C(X, Y)$ is sequential?

Let $$X$$ be a topological space, $$(Y, d)$$ a metric space and $$C(X, Y)$$ the space of continuous maps with the topology of compact convergence.

Question: What is a minimal topological condition on $$X$$ such that $$C(X, Y)$$ is a sequential space?

The motivation of the question is the following:

Let $$(f_n)$$ be a sequence in $$C(X, Y)$$ that is equicontinuous and $$\overline{ \{ f_n(x) : n \in \mathbb{N} \} }$$ is compact in $$(Y, d)$$ for every $$x \in X$$. By Ascoli theorem, the closure $$\overline{ \{ f_n : n \in \mathbb{N} \} }$$ in $$C(X, Y)$$ is compact. Unfortunately, in general $$(f_n)$$ may not have any convergent subsequence (in the topology of compact convergence).

To guarantee a convergent subsequence, it is enough for $$C(X, Y)$$ to be a sequential space. Then $$\overline{ \{ f_n : n \in \mathbb{N} \} }$$ would be a compact subspace that is also sequential, which implies that it is sequentially compact.

It is well known that if $$X$$ is hemicompact, then $$C(X, Y)$$ is metrizable. Of course $$C(X, Y)$$ would be sequential in this case. But the hemicompactness of $$X$$ is too strong a condition. We only need $$C(X, Y)$$ to be sequential, and metrizability is a big overkill.

On the other hand, it is not enough for $$X$$ to be a locally compact metric space. See this Math.SE page for a counter-example.

• Here are some remarks on the special case that $Y=\mathbb{R}$ and $X$ is Tychonoff. Then $C(X)$ is (strongly) Fréchet-Urysohn iff it is sequential iff it is compactly generated. On the other hand, if $X$ is first-countable, then $C(X)$ is Fréchet-Urysohn iff $X$ is hemicompact. In general I think even the case that $Y=$ seperable metric is not too well understood. Jun 23 at 11:11
• The reference for the above is a paper by E.G. Pytkeev. A recent refernce to his paper is from Topological properties of some function spaces by Gabriyelyana and Osipov, which contains some statements you might be interested in. Jun 23 at 11:23