Descriptive Topology in non-Archimedean Function Spaces Cp(X, K). Part I

Abstract

Let $K$ be a non-archimedean field and let $X$ be an ultraregular space. We study the non-archimedean locally convex space $C_p(X;K)$ of all $K$-valued continuous functions on $X$ endowed with the pointwise topology. We show that $K$ is spherically complete if and only if every polar metrizable locally convex space $E$ over $K$ is weakly angelic. This extends a result of Kiyosawa - Schikhof for polar Banach spaces. For any compact ultraregular space $X$ we prove that $C_p(X;K)$ is Frechet-Urysohn if and only if $X$ is scattered (a non-archimedean variant of Gerlits - Pytkeev's result). If $K$ is locally compact we show the following: (1) For any ultraregular space $X$ the space $C_p(X;K)$ is K-analytic if and only if it has a compact resolution (a non-archimedean variant of Tkachuk's theorem); (2) For any ultrametrizable space $X$ the space $C_p(X;K)$ is analytic if and only if $X$ is $\sigma$-compact (a non-archimedean variant of Christensen's theorem).