The combinatorics of open covers (Q2874872)

This is an excellent survey of recent progress in selection principles theory and its relationships with various fields of mathematics: game theory, Ramsey theory, function spaces, cardinal invariants, dimension theory and so on. The paper is divided into seven sections, but the three main parts are sections 4, 5, 6. In Section 4 the authors discuss the Rothberger-type selection properties (Rothberger spaces and groups, Gerlits-Nagy spaces, countable fan tightness, Frechét-Urysohn spaces, strong selective separability) described in a general form as \(S_1(\mathcal A, \mathcal B)\) properties for several classes \(\mathcal A\) and \(\mathcal B\). Game-theoretic and Ramsey-theoretic characterizations of some of these properties are given. In particular, classes of compact Rothberger spaces that arise in function spaces theory are considered, as well as the cardinality of Rothberger spaces.NEWLINENEWLINEIn Section 5 a similar scenario is applied to the Menger-type selection properties \(S_{fin}(\mathcal A,\mathcal B)\). Results related to Menger spaces, Hurewicz spaces, countable fan tightness, selective separability are reported. Section 6 deals with the principle denoted \(S_c(\mathcal A,\mathcal B)\), whose prototype is the selective screenability, and which have deep connections with (weakly and strongly) infinite dimensional spacesNEWLINENEWLINESome parts of selection principles theory (for example, uniform or star selection properties) have not been considered here; it is clear that one such survey cannot cover all directions of investigations in this quickly growing field of mathematics. However, the authors give a long list of references where the interested reader can find additional information.NEWLINENEWLINEFor the entire collection see [Zbl 1282.54001].