This is an unexpected example of an application of the principles and methods of general topology to set theory. Namely, the main importance of this paper is the discovery that the Menger's covering property of filters characterizes a classical combinatorial property of the Mathias forcing associated to it. In a series of papers, the emerging prominent topologist Michael Hrusak and others have intensively studied a type of filters with this property, introduced by Canjar. In our paper we proved that a filter has Canjar's property if and only if it is Menger in the standard topology. This made the wide body of knowledge on Menger's property applicable to Canjar filters. In particular, many earlier results immediately follow from our characterization.
COBISS.SI-ID: 17556057
One of the major open problems in set theory of the reals is the existence of a non-meager P-filter. Thus it is important to improve our understanding of such filters, e.g., by establishing additional properties thereof, towards the solution of the problem mentioned above. In this paper several topological and combinatorial conditions are given which, for a filter on w, are equivalent to being a non-meager P-filter. In particular, it is shown that a filter is countable dense homogeneous if and only if it is a non-meager P-filter. Here a filter is identified with a subspace of 2^w through characteristic functions. Along the way, a result of Miller about P-points is generalized to non-meager P-filters, and a new proof of a result of Marciszewski is given. One of the ingredients of the main proof is a deep theorem of Hernandez-Gutierrez and Hrusak, and two of their questions are answered. The main result of this paper also resolves several issues raised by Medini and Milovich. Furthermore, it is also shown that the statement “Every non-meager filter contains a non-meager P-subfilter” is independent of ZFC (more precisely, it holds in the Miller model and its negation is a consequence of the Jensen's diamond principle). It is worth mentioning here that it follows from results of Hrusak and van Mill that in the Miller model, a filter has less than c types of countable dense subsets if and only if it is a non-meager P-filter. In particular, in this model there exists an ultrafilter with c types of countable dense subsets. In the paper under discussion it is also shown that such an ultrafilter exists under a small portion of Martin's Axiom. The results of this paper have received a lot of interest from set-theorists and topologists. To mention some, recently J. Brendle, B. Farkas, and J. Verner started joint collaboration on ultrafilters containing no non-meager towers, a topic originating in our work.
COBISS.SI-ID: 17439577
It is proved that for each meager relation on a Polish space X there is a nowhere meager subspace Y of X which is free with respect to this relation, which means that no pair of elements of Y lies in the relation. The problem of finding “large” free sets Y for certain “small” relations is a classical topic which has attracted many prominent researchers like Mycielski, Kuratowski, Solecki, and Pawlikowski. The main result of this paper has immediately found applications in the study of so-called supercompact spaces by Banakh and his collaborators. The results obtained in this paper have received a lot of interest from set theorists and topologists. To mention some, recently J. Brendle and B. Farkas started joint collaboration on free sets with respect to relations which lie in other ideals with descriptive nature, e.g., in the ideal of Lebesgue null sets.
COBISS.SI-ID: 17310809