Descriptive Classes of Sets in Nonseparable Spaces

The classical descriptive set theory is devoted to the study of descriptive classes of sets in Polish spaces. There are many deep results which found applications, e.g., in real analysis or functional analysis. Several books and survey papers contain essential parts of this theory, see [49], [2], [48], [54], [63]. One of the key tools for the study of Borel sets in Polish spaces are analytic spaces. There are several attempts to transfer parts of the theory to more general spaces. Among them two seem to form a satisfactorily rich theory (K-analytic spaces and absolute Souslin sets in metrizable spaces) and, in particular the first one, found interesting applications in the study of nonseparable Banach spaces and weak topologies on them. As representative surveys may serve [8] and [44]. Since these results enable to study just Lindelöf or metrizable topologies, some further extensions, inspired by the previous ones, were made by R.W. Hansell. There is a survey paper on generalized analytic spaces (on the ”descriptive topology”), which points out some of key results, written by Hansell, see [22], cf. also [28], [32]. We start our survey by a choice of facts and problems concerning the descriptive classes of sets in metrizable spaces. We point out in particular some recent results concerning the mappings which preserve some descriptive properties and we list several still open problems which seem to be quite essential for the full understanding of the theory. Many of them are closely related and have a positive answer under an axiom of W.G.Fleissner (which is consistent with ZFC provided the existence of a supercompact cardinal is consistent). Then we are going to discuss descriptive classes of sets in Tychonoff spaces. An important feature is that there are several natural notions of analyticity. They define classes of generalized analytic spaces which are ordered by the inclusion, but there is no unified attitude to all of them. The classes coincide within metrizable spaces and some results may be received repeating the methods used for metrizable spaces. However, many problems are specific. We recall some old and some more recent results, and we point out some open questions which seem to make the extension of the classical descriptive set theory and its applications to functional analysis not yet sufficiently complete. We are aware that we cannot cover all related results and problems. For example, we mention only a few results concerning the Borel classes of sets and mappings, and we do so mainly when applying them to results on the σ-algebra of all Borel