Variations of classical selection principles: an overview

Abstract

The paper is an overview of selected results on weaker forms of classical selection principles of Menger, Hurewicz, Rothberger and Gerlits-Nagy obtained in the last few years. We review some very recent results on star selection principles, an important subfield of selection principles theory (including boundedness properties in function spaces considered as uniform spaces and topological groups), ideal versions of Hurewicz-type properties, selection principles called weak, almost, quasi, and combinatorics of covers by generalized open sets. A number of examples which clarify relationships among properties we consider is given, and some open questions are posed.

Mathematics Subject Classification (2010): Primary: 54D20; Secondary: 05C55, 54B99, 54C99, 54D10, 54E15, 91A44.

Keywords: Selection principles, weak (almost, quasi, set) selection principles, game theory, Ramsey theory, uniform space