## 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.

### MSC:

 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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