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Descriptive sets and the topology of nonseparable Banach spaces. (English) Zbl 0982.46012
The object of this paper is to introduce and investigate several classes of non-separable Banach spaces whose weak topologies possess a certain type of network, similar to the types of networks investigated in the study of generalized metric spaces. The classes studied in the paper provide a natural extension of the class of K-analytic and countably determined Banach spaces, but need not be Lindelöf in the weak topology. A descriptive topological space \(X\) is defined as an image of a complete metric space \(T\) under a continuous surjection \(f\) such that, whenever \(\{ E_\lambda :\lambda \in \Lambda\}\) is a relatively discrete family in \(T\), then \(\{ f(E_\lambda):\lambda\in\Lambda\}\) is \(\sigma\)-relatively discretely decomposable in \(X\). It is shown (among others) that a Banach space (that is, the space endowed with its weak topology) is descriptive precisely when it has a \(\sigma\)-relatively discrete network. In 2000 Oncina proved that the latter class coincide with the Banach spaces having countable cover by sets of small local diameter (or JNR). A Hausdorff topological space \(X\) is said to be almost descriptive, if there is a complete metric space \(T\) and a continuous surjection \(f:T\rightarrow X\) such that, whenever \(\{ E_a :a\in A\}\) is a scattered family in \(T\), \(\{ f(E_a) :a\in A\}\) is point-countable and has a \(\sigma\)-scattered base. It is proved that \(\sigma\)-fragmented and almost descriptive Banach spaces coincide (equivalently, they admit a \(\sigma\)-scattered network). Topological and embedding properties of (almost) descriptive (and of (almost) K-descriptive) spaces are discussed in considerable depth.

46B20 Geometry and structure of normed linear spaces
54H05 Descriptive set theory (topological aspects of Borel, analytic, projective, etc. sets)
54E99 Topological spaces with richer structures
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