Descriptive topology in non-Archimedean function spaces \(C_{p}(X, \mathbb K)\). I (Q2918784)

Let \(K\) be a non-Archimedean non-trivially valued complete field. ln the paper under review, the authors show that \(K\) is spherically complete if and only if every polar metrizable locally \(K\)-convex space is weakly angelic, thereby extending a result of \textit{T. Kiyosawa} and \textit{W. H. Schikhof} [Int. J. Math. Math. Sci. 19, No. 4, 637--642 (1996; Zbl 0856.46051)].NEWLINENEWLINE The main purpose of the present work is the study of certain properties of the \(K\)-vector space \(C(X,K)\) of continuous functions from the ultraregular space \(X\) into \(K\) endowed with the locally \(K\)-convex topology of simple convergence.NEWLINENEWLINE More precisely, by using arguments of \textit{V. V. Tkachuk} [Acta Math. Hung. 107, No. 4, 253--265 (2005; Zbl 1081.54012)], the authors prove that, if \(K\) is locally compact, then \(C(X,K)\) is analytic if and only if it admits a compact resolution. They also prove that, if \(X\) is compact, then \(C(X,K)\) is Fréchet-Urysohn if and only if \(X\) is scattered, a non-Archimedean version of a result of Gerlits and Pytkeev (see [\textit{A. V. Arkhangel'skij}, Topological function spaces. Mathematics and its Applications, Soviet Series 78. Dordrecht etc.: Kluwer (1992; Zbl 0758.46026)], Theorem III.1.2).NEWLINENEWLINE Moreover, they show that, if \(K\) is locally compact and \(X\) is ultrametrizable, then \(C(X,K)\) is analytic if and only if \(X\) is \(\sigma\)-compact, a non-Archimedean version of a result of \textit{J. P. R. Christensen} [North-Holland Mathematics Studies 10. Amsterdam-London: North-Holland; New York: American Elsevier (1974; Zbl 0273.28001)]. Pertinent applications of the above-mentioned theorems are also presented in the paper.