We present a ZFC construction of a non-meager filter which fails to be countable dense homogeneous. This answers a question of Hernández-Gutiérrez and Hrušák. The method of the proof also allows us to obtain for any n \in \omega \cup \{ \infty \} an n-dimensional metrizable Baire topological group which is strongly locally homogeneous but not countable dense homogeneous.
COBISS.SI-ID: 16962905
We prove that under certain set-theoretic assumptions every productively Lindelöf space has the Hurewicz covering property, thus improving upon some earlier results of Aurichi and Tall. This paper was among the top downloaded papers in 2014 for the journal Topology and its Applications (Elsevier).
COBISS.SI-ID: 16975961
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
In this paper we study the fundamental group of inverse limits, obtained by upper semi-continuous set valued functions. We present a number of crucial examples which demonstrate the technical difficulties, related to the control of the fundamental group in the inverse limit. Furthermore, these examples realize some important groups as the fundamental groups of inverse limits: free groups and the Hawaiian Earring group. On the other hand, we introduce the right shift of a loop in the inverse limit and prove that the fundamental group of an inverse limit, which is a one-dimensional Peano continuum, is often trivial or uncountable.
COBISS.SI-ID: 17610585
We 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 collaboration with us 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