Let $X$ be a compact metric space and $Y$ a stratifiable space. By $C(X, Y)$, we denote the space of continuous maps from $X$ to $Y$ with the compact-open topology. In general, $C(X, Y)$ is not stratifiable. In this paper, we show that $C(X, Y)$ is stratifiable if $Y$ satisfies the condition given by Mizokami [Mi]. And we construct a stratifiable space $Y$ such that $C(X, Y)$ is not stratifiable even if $X$ is countable and compact.
"Function spaces which are stratifiable." Tsukuba J. Math. 18 (2) 505 - 517, December 1994. https://doi.org/10.21099/tkbjm/1496162617