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A characterization of Valdivia compact spaces. (English) Zbl 0949.46004

We characterize Valdivia compact spaces \(K\) in terms of \(C(K)\) endowed with a topology introduced by M. Valdivia [Collect. Math. 42, No. 3, 265-285 (1991; Zbl 0788.47024)]. This generalizes R. Pol’s characterization of Corson compact spaces [“On pointwise and weak topology on function spaces”, preprint 4/84, Warszaw Univ., 1984; see 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 S. Argyros, S. Mercourakis and 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.
Reviewer: O.Kalenda (Praha)

MSC:

46A50 Compactness in topological linear spaces; angelic spaces, etc.
46E10 Topological linear spaces of continuous, differentiable or analytic functions
54C10 Special maps on topological spaces (open, closed, perfect, etc.)
54C35 Function spaces in general topology
54D30 Compactness
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