Descriptive sets and the topology of nonseparable Banach spaces (Q2756974)

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.