Topological properties of function spaces over ordinals

Abstract

A topological space X is said to be an Ascoli space if any compact subset $\mathcal {K}$ of $C_k(Y)$ is evenly continuous. This definition is motivated by the classical Ascoli theorem. We study the $k_\mathbb {R}$-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 $k_\mathbb {R}$-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 $k_\mathbb {R}$-space, namely $C_p(\omega _1)$. We show that $C_k(\kappa )$ is Ascoli iff $\mathrm {cf}(\kappa )$ is countable iff $C_k(\kappa )$ is metrizable.