Construções genéricas de espaços de Asplund C(K)

Abstract

In this work we consider a method of generic constructions of compact scattered nonmetrizable spaces developed by Baumgartner, Shelah, Rabus, Juhasz and Soukup. We introduce new techniques and obtain new applications both relevant to topology of compact spaces and the geometry of Banach spaces of continuous functions. The new techniques concern new amalgamations of conditions of forcing which add the dispersed spaces as well as the generalizations of arguments of the above-mentioned authors from points of a compact space K to Radon measures on K. As applications we obtain two compact scattered spaces K1 and K2 with the properties below. K1 is a hereditarily separable space of weight א1 such that C(K1) has property (C) of Corson and does not have property (E) of Efremov. K2 is the first (consistent) example of a compact scattered space which is hereditarily separable and whose height is ω2. It follows that its hereditary Lindelof degree is א2, showing the consistency of hL(K) 6≤ hd(K)+ for compact spaces K. C(K2) is the first consistent example of a Banach space of density א2 without uncountable biorthogonal systems.