Topological properties of function spaces over ordinals

A topological space $X$ is said to be an Ascoli space if any compact subset $K$ of $C_k(X)$ is evenly continuous. This definition is motivated by the classical Ascoli theorem. We study the $k_R$-property and the Ascoli property of $C_p(\kappa)$ and $C_k(\kappa)$ over ordinals $\kappa$. We prove that $C_p(\kappa)$ is always an Ascoli space, while $C_p(\kappa)$ is a $k_R$-space iff the cofinality of $\kappa$ is countable. In particular, this provides the first $C_p$-example of an Ascoli space which is not a $k_R$-space, namely $C_p(\omega_1)$. We show that $C_k(\kappa)$ is Ascoli iff $cf(\kappa)$ is countable iff $C_k(\kappa)$ is metrizable.