A characterization of Valdivia compact spaces
type: Item
instance of: scholarly article
title: A characterization of Valdivia compact spaces
zbMATH Open document ID: 0949.46004
published in: Collectanea Mathematica
publication date: 2000-12-03T00:00:00Z
full work available at URL: https://eudml.org/doc/41496
review text: We characterize Valdivia compact spaces \(K\) in terms of \(C(K)\) endowed with a topology introduced by \textit{M. Valdivia} [Collect. Math. 42, No. 3, 265-285 (1991; Zbl 0788.47024)]. This generalizes \textit{R. Pol}'s characterization of Corson compact spaces [``On pointwise and weak topology on function spaces'', preprint 4/84, Warszaw Univ., 1984; see \textit{A. V. Arkhangel'skij}, ``Topological functions spaces'', Moscow (in Russian) (1989; Zbl 0781.54014), Dordrecht (Engl. transl.) (1992; Zbl 0758.46026)]. An analogous characterization is proved for Banach spaces with Valdivia dual unit ball. This generalizes the characterization of weakly Lindelöf determined Banach spaces given by \textit{S. Argyros}, \textit{S. Mercourakis} and \textit{S. Negrepontis} [Stud. Math. 89, No. 3, 197-229 (1988; Zbl 0656.46014)]. Further, we study duality, products and open continuous images of Valdivia compact spaces. We prove in particular that the dual unit ball of \(C(K)\) is Valdivia whenever \(K\) is Valdivia and that the converse holds whenever \(K\) has a dense set of \(G_\delta\) points. Another result is that any open continuous image of a Valdivia compact space with a dense set of \(G_\delta\) points is again Valdivia.
Mathematics Subject Classification ID: 46A50
Mathematics Subject Classification ID: 46E10
Mathematics Subject Classification ID: 54C10
Mathematics Subject Classification ID: 54C35
Mathematics Subject Classification ID: 54D30
zbMATH DE Number: 1454852
zbMATH Keywords: primarily Lindelöf space
zbMATH Keywords: Valdivia dual unit ball
zbMATH Keywords: spacewith a dense set of \(G_\delta\) points
zbMATH Keywords: Valdivia compact spaces
zbMATH Keywords: Corson compact spaces
author: Ondřej F. K. Kalenda
reviewed by: Ondřej F. K. Kalenda
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1454852