Descriptive Topology in non-Archimedean Function Spaces Cp(X, K). Part I
Loading...
Date
2012
Authors
Advisor
Editor
Journal Title
Journal ISSN
Volume Title
Publisher
Title alternative
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).
Description
Sponsor
The National Centre of Science, Poland (grant no. N N201 605340)
Keywords
Citation
Bull. London Math. Soc., 44(2012), 899-912