Game theory of generalized selection hypotheses

Abstract: This talk is an opportunity to get acquainted with the basic definitions and some elementary techniques used in the new field often called "selection principles". We will do that by introducing one aspect of the field only: That of topological games. Motivated by a question of Iliadis, Scheepers' prototypes of topological diagonalizations are extended to the case that the types of covers in the sequence to be diagonalized can vary. A known example of such a property which cannot be expressed by Scheepers' prototypes is the Galvin-Miller strong gamma-property. Whereas this property is strictly stronger than the Gerlits-Nagy gamma property, the corresponding strong notions for the Menger, Hurewicz, Rothberger, Gerlits-Nagy (*), Arkhangel'skii and Sakai properties are equivalent to the original ones. (All these properties will be defined in the talk.) We give new game theoretic characterizations for most of these properties, and pose several interesting open problems. No background is assumed and the talk will be aimed at a general mathematical audience. All needed definitions will be provided, and most of the proofs will be given or at least sketched. As much time as required will be dedicated to answering questions of the audience. The talk is an extended version of a planned talk in the upcoming Bonn Conference on infinite games.