Descriptive topology in non-archimedean function spaces C _p (X , 핂). Part I

Abstract

Let K be a non-archimedean field and let X be an ultraregular space. We study the nonarchimedean 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 Fréchet-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 σ-compact (a non-archimedean variant of Christensen's theorem).