Forcing, fusion, and combinatorics of open covers

The main objective of the project is the interplay between combinatorial covering properties and forcing. Combinatorial covering properties which are often also called selection principles arose from the study of special sets of reals. These resolved many classical questions in general topology and measure theory. As a result, information about them is included in standard topology textbooks, such as Kuratowskis Topology. This theory is a remarkable and deep part of general topology, bordering with set theory and measure theory. The theory of selection principles, which originated in works of Scheepers 20 years ago, provides a modern treatment to that of special sets of reals, and incorporates the latter in a broader, unified and far-reaching framework. The uniform treatment given by selection principles to the vast variety of fundamental mathematical properties not only made these studies more aesthetically pleasing, but also brought new perspectives into them, focused attention on essential parts of classical problems, and led to their satisfactory solution. Another main ingredient of the project is forcing. It entered logic through the seminal work of Paul Cohen in 1964, where he proved that the Continuum Hypothesis does not follow from the usual axioms of mathematics. This method turned out to be a remarkably general technique for producing new models and consistency results. The main idea of forcing is to extend a universe by adjoining a new (generic) element in order to obtain a larger universe called a generic extension. Since then there has been plenty of striking applications in various areas of mathematics including algebra, analysis, measure theory, topology, etc. However, until recently, there seem to have been only isolated application of forcing to selection principles. We expect that nevertheless this is a rich area which is to be explored, and we have already started to do this by proving certain preservation by products results in the classical models of Miller and Laver. These results seem to be an instance of a more general phenomenon, namely that fusion affect the behavior of selection principles. Regarding the application of combinatorial covering properties to forcing, it seems to be encountered as late as in 2015 in our paper with Chodounsky. Here we plan to work along this line of research further. In our collaboration the Austrian and Slovenian sides contribute their set-theoretic and topological expertise, respectively, both of which are of crucial importance for this project. In addition to solution of open problems we plan to contribute to a deeper understanding of the methods used. In particular, we plan to detect and isolate features of forcing and selection principles useful for the studies mentioned above.

The aim of this project was an investigation of the interaction of combinatorial covering properties with forcing. Combinatorial covering properties, also known in the literature as selection principles, have been developed in the course of the investigation of so-called special subsets of the real line. With the help of these subsets numerous old problems of topology and measure theory could be solved. As a result they are treated in standard textbooks on topology, e.g. in Kuratowski`s book. Over the years, the study of selection principles has become a remarkably deep part of general topology. Today, numerous connections of selection principles to set theory and measure theory are known. The second technique used in the project is forcing, which was developed by P. Cohen in 1964 to show that the famous continuum hypothesis is independent of the classical ZFC axioms. Subsequently, the forcing method has proved exceedingly useful in constructing new models of ZFC and obtaining a variety of independence results. In recent decades, a variety of interesting applications of the forcing method to other areas of mathematics (algebra, analysis, measure theory, topology, etc.) have been found. Until recently, however, there have been only isolated applications of the forcing method in the study of selection principles. Our results, obtained in the course of work on the project, show that there is a rich theory of such applications. Our results in this direction include preservation by products in the classical models of Laver and Miller. Some of these preservation results also involve local properties of function spaces. Our new results have also confirmed our expectation of a more general phenomenon, namely that so-called fusion affects the behaviour of selection principles in corresponding forcing extensions. Furthermore, we were able to apply ideas from our study of selection principles to forcing constructions. These applications build on our 2015 results, especially when it goes about the destruction of maximal almost disjoint families. In our collaboration, both the Austrian and Slovenian sides contributed their expertise from set theory and topology. Both areas were of fundamental importance for the project. Besides solving open problems, we also made progress towards isolating relevant (for the results described above) properties of the forcing posets and selection principles.