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.