Descriptive topology. (English) Zbl 0805.54036

Hušek, Miroslav (ed.) et al., Recent progress in general topology. Papers from the Prague Toposym 1991, held in Prague, Czechoslovakia, Aug. 19-23, 1991. Amsterdam: North-Holland. 275-315 (1992).
From the introduction: “We have endeavored to first give a brief overview of the progressive development of the concept of an analytic space, from its classical setting to its most recent form, in order to establish a framework and standard with which to analyze and compare the status of recent progress on this subject. In Section 2 we give the classical definition of an analytic set and summarize four theorems that are at the heart of the subject. The general “separable” theory of \(K\)- analytic sets is reviewed in Section 3, particularly the role played by upper semi-continuous compact-valued maps. The non-separable theory is considered in Section 4, where we highlight the importance of the role played by \(\sigma\)-discrete collections in place of countable collections in the classical theory. In Section 5 we initiate our study of recent progress made in descriptive topology by considering in detail the class of Čech-analytic spaces. Although this class of spaces has been utilized in a number of recent papers, we indicate some outstanding problems in the study and some apparent shortcomings. Our final and most detailed exposition is given in Section 6 where we discuss two classes of \(P\)-\(K\)-analytic spaces. Here \(P\) denotes a property of disjoint collections of subsets of a topological space similar to the property of discreteness which was employed so successfully in the non-separable theory. Although it would be possible to discuss a purely abstract form of this concept, we deal here primarily with the case when \(P\) stands for a “scattered” or an “isolated” collection. Some applications are discussed and others are referenced, and several open problems are indicated.” The “References” comprise 61 titles.
For the entire collection see [Zbl 0782.00072].


54H05 Descriptive set theory (topological aspects of Borel, analytic, projective, etc. sets)
54-02 Research exposition (monographs, survey articles) pertaining to general topology