The topology of filters

This research project is in general topology, and it is related to set theory in the following ways. -Use set-theoretic axioms (like Martin`s Axiom or Diamond) or assumptions on cardinal invariants to prove consistency or independence results about topological statements. -Study combinatorial objects on omega (especially filters) from the topological point of view. -Make use of/investigate topological properties of definable sets (Borel, analytic, coanalytic, and so on). Filters are classical objects in mathematics, and semifilters are a natural generalization of this notion which has recently found several applications (for example in the theory of selection principles). Both filters and semifilters can be given a natural structure of topological space (with the subspace topology that they inherit from the Cantor set), and their combinatorial structure turns out to significantly constrain their topology, especially in terms of homogeneity-type properties. A large part of our project consists in extending the remarkable results of Fons van Engelen on Borel filters and Borel zero-dimensional homogeneous spaces beyond the Borel realm (assuming suitable Determinacy axioms). This possibility has been opened up by recent results of Fournier that extend Louveau`s deep work on the Borel Wadge classes, which is a crucial ingredient in van Engelen`s work. We also plan to do the same for recent results of the applicant on Borel semifilters. Furthermore, we will investigate a conjecture due to van Engelen on the relationship between filters and topological groups. Several questions in our project are related to the perfect set property, which is a classical notion in descriptive set theory. Here, however, we will study it from a different point of view, which is conceptually new. In particular, inspired by results of Miller, we will investigate the relations between various perfect set type- properties (such as Marczewski measurability) for ultrafilters. Finally, our project involves the notion of countable dense homogeneity, which traces back to the work of Georg Cantor himself. In particular, we would like to characterize the semifilters that have this property, and the zero- dimensional infinite powers that have this property (the second problem is due to Fitzpatrick and Zhou).

The main focus of our research has been on Wadge theory, which gives a systematic analysis of the complexity of the subsets of a space. Starting with the work of F. van Engelen, this has proved to be an invaluable tool in the study of homogeneous spaces (including filters and semifilters). In joint works with R. Carroy, S. Müller and Z. Vidnynszky, we have both established purely Wadge-theoretic results (heavily based on unpublished work of A. Louveau) and given applications to strong homogeneity and s-homogeneity. While van Engelen`s work was limited to the Borel context, we obtained results for all spaces, under AD (Axiom of Determinacy). We believe that our work will make it possible to give a complete classification of the zero-dimensional homogeneous spaces, and of filters and semifilters on under AD.