General topology and set-theoretic methods

The main theme of the proposed research is the application of set-theoretic methods such as forcing, elementary submodels, cardinal characteristics, and their syntheses with topological methods like selections of set-valued maps, to questions in general topology. For instance, we plan to investigate whether regular Lindelöf spaces and, more generally, paracompact spaces are D-spaces by using elementary submodels and set-valued maps. We also expect a significant progress in understanding productively Lindelöf spaces in various models of set theory and their relations to D-spaces, which may have a strong impact on the study of Michael spaces. In addition to solutions of some of the problems mentioned below we plan to contribute to a deeper understanding of the above- mentioned methods. The notion of a D-space was introduced by van Douwen and Pfeffer in 1979 and since then it has been one of the main concepts in general topology. As it was noticed by Eisworth in 2007, one of the main objectives of the study of D-spaces is to gain a better understanding of the relationship between covering properties and the state of being a D-space. In particular, the following problems are open: Is every regular Lindelöf space X a D-space? Is every paracompact space a D-space? First remarkable progress in this direction has recently been made by L. Aurichi, who proved that every space with Menger`s covering property is a D-space. In a joint work of Repovš and Zdomskyy we isolated and extended the combinatorial core of the Aurichi`s proof and established that under Martin`s Axiom, every paracompact space of size is a D-space, thus solving a problem of G. Gruenhage. One of the aims of our project is to strengthen the abovementioned result, the ultimate goal being to avoid the additional set-theoretic assumptions, i.e., to answer the second problem mentioned above. Besides the use of appropriate games as before, we plan to use the theory of set- valued maps and their selections, where D. Repovš has an outstanding expertise. This approach is based on our observation that there are many natural set-valued maps with strong topological properties related to a neighborhood assignment N capturing all information about it. We expect that elementary submodels could be very useful here. Another related problem is: Is there a Lindelöf space X whose product with the space of irrationals is not Lindelöf? Spaces X giving the affirmative solutions to Problem 3 are called Michael spaces. There are many consistent examples of Michael spaces. However, the complete solution of this problem seems to be beyond the reach of currently known methods, and hence it is natural to approach it in "small" steps. One of the natural approaches implicitly suggested by F. Tall in his recent papers is to analyze productively Lindelöf sets of reals. It has been recently proved by Repovš and Zdomskyy that if there exists a Michael space, then every productively Lindelöf space has Menger`s property, and therefore is a D-space. This result seems to unveil a previously hidden connection between problems mentioned above, and we plan to explore it further. Another intriguing area is the behavior of productively Lindelöf sets of reals under the negation of the Continuum Hypothesis. We plan to check whether all such sets are -compact in the Miller and Sacks models of set theory. We expect affirmative answer for projective sets of reals.

The main theme of the project was the application of set-theoretic methods such as forcing, elementary submodels, cardinal characteristics, and their syntheses with topological methods like set-valued maps, to questions in general topology. Our most important achievements are related to preservation by products of various covering properties of topological spaces. For instance, we have proved with Andrea Medini that under Continuum Hypothesis, every productively Lindelöf space of countable tightness has also Lindelöf countable power. This generalized several results of Tall, Tsaban, and others and is nowadays the best known approximation to the answer of the following famous question of E. Michael: Does every productively Lindelöf space have Lindelöf countable power? Our proof employs set-valued maps and elementary submodels of the ambient universe. In our joint work with Leandro Aurichi we have succeeded to characterize topological spaces whose product with every Lindelöf space is Lindelöf in terms of their internal properties, which answered a classical problem attributed to Tamano. The main idea of that charactrization was to analyze an auxiliary space whose points are certain families of open sets with suitable combinatorial structure. This allowed us to improve an old result of Alster characterizing productively Lindelöf spaces with the smallest possible uncountable weight under Continuum Hypothesis. Unlike in the Alsters result, in our improvement we do not make any restrictions on the size of Continuum. Another successful line of research in the framework of this project was interplay between combinatorial covering properties and forcing. Regarding the application of combinatorial covering properties to forcing, it seems to be encountered as late as in 2015, in our joint paper with Chodounsky and Repovs devoted to filters on the natural numbers, forcing notions associated to them, cardinal characteristics, and preservation of boundedness type properties of the ground model reals. In particularly, we proved that the above mentioned preservation for the Mathias forcing associated to a filter has an unexpected topological nature, which allowed to answer numerous questions of Brendle, Hrusak, Minami, and others. Speaking of the other direction in this research, until recently, there seem to have been only isolated applications of forcing to combinatorial covering properties, which are direct in the sense that one deals actually with the forcing poset, and not with certain (in)equalities between cardinal characteristics in the corresponding forcing extension. In our work we have indicated that this might be a rich area which is to be explored, by proving preservation results for the covering properties of Menger and Hurewicz in classical models constructed by Miller and Laver, respectively. In all of these works a special attention has been paid to the further development of the methods we used.