Topological properties of function spaces over ordinal spaces

Saak Gabriyelyan, Jan Grebik, Jerzy Kakol, Lyubomyr Zdomskyy

Published 2016-06-13Version 1

Motivated by the classical Ascoli theorem, a topological space $X$ is said to be an Ascoli space if any compact subset $\mathcal K$ of $C_k(Y)$ is evenly continuous. We study the $k_{\mathbb R}$-property and the Ascoli property of $C_p(\kappa)$ and $C_k(\kappa)$ over ordinal spaces $\kappa=[0,\kappa)$. 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.