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Topological properties of Banach spaces. (English) Zbl 0793.54026

The paper investigates \(\sigma\)-fragmentability of non-compact subsets of Banach spaces and, indeed, of Banach spaces themselves. A subset \(Z\) of a Banach space \(X\) is fragmented by the norm if, for every \(\varepsilon > 0\) and every non-empty subset \(E\) of \(Z\), there is a non-empty relatively weak open subset of \(E\) with norm diameter less than \(\varepsilon\). Given a topological space \(Z\) and a metric \(\rho\) on \(Z\), the space \(Z\) is \(\sigma\)-fragmented by \(\rho\), if, for each \(\varepsilon > 0\), \(Z\) can be expressed as a countable union of sets, each with the property that each non-empty subset has a non-empty relatively open subset of \(\rho\)- diameter less than \(\varepsilon\). Further, although the terminology may not seem very apt, the space \(Z\) is called \(\sigma\)-separable with respect to \(\rho\), if, for each \(\varepsilon > 0\), \(Z\) can be expressed as a countable union of sets, each with the property that each non-empty subset has a non-empty relatively open subset that can be covered by a countable family of sets each of \(\rho\)-diameter less than \(\varepsilon\). These notions coincide with the notion of fragmentability when \(Z\) is a compact subset of a Banach space and \(\rho\) is the norm metric. The main theorem of the paper needs the notion of Čech-analytic spaces in the sense of D. H. Fremlin, the definition of which is too involved to be cited here. The main theorem is: Let \(\rho\) be a lower semi-continuous metric on a Hausdorff space \(Z\). Then the implications \((\text{a}) \Leftrightarrow (\text{b}) \Rightarrow (\text{c}) \Rightarrow (\text{d})\) hold among the following conditions. Further, if \(Z\) is Čech- analytic, then all the conditions are equivalent: (a) \(Z\) is \(\sigma\)- fragmented by \(\rho\). (b) \(Z\) is \(\sigma\)-separable with respect to \(\rho\). (c) Each compact subset of \(Z\) is fragmented by \(\rho\). (d) For no \(\varepsilon > 0\), does \(Z\) contain a compact set \(H\) that admits a continuous map onto the Cantor set \(2^ N\) with inverse images of distinct points of \(2^ N\) separated by \(\rho\)-distance at least \(\varepsilon\). The difficult proof that \((\text{c}) \Rightarrow (\text{a})\), when \(Z\) is Čech-analytic, follows the general line of M. Souslin’s (1917) proof that an uncountable analytic subset of Euclidean space contains a Cantor set; more precisely, it follows a generalization of Souslin’s result and makes essential use of a kernel argument similar to one due to A. H. Stone [Trans. Am. Math. Soc. 107, 58-70 (1963; Zbl 0114.386). It follows from the main theorem that a dual Banach space \(X^*\) with its \(\text{weak}^*\) topology is \(\sigma\)-fragmented by its norm-metric if, and only if, \(X\) is an Asplund space. Further all Banach spaces having Kadec norms are \(\sigma\)- fragmented, but the space \(\ell^ \infty\) is not \(\sigma\)-fragmented. The paper contains a number of other interesting results. For example, it is shown that the notion of \(\sigma\)-fragmentability reduces to fragmentability if the space in question is hereditarily Baire. Examples are given to illustrate various points of the theory. Finally an appendix contains a proof of Fremlin’s unpublished results on Čech-analytic spaces.

MSC:

54E99 Topological spaces with richer structures
54H05 Descriptive set theory (topological aspects of Borel, analytic, projective, etc. sets)
46B22 Radon-Nikodým, Kreĭn-Milman and related properties
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