Set-theoretic aspects of topological selections

Defining a mathematical property, concrete examples of objects with this property are needed. How to construct objects with a given property? Relativity in mathematics. Could it be that two properties are different in one world, but they are the same in another one? In mathematics, any statement seems to be true or not, but sometimes it is undecidable, and this fact can be proved. Do there exist objects with a given property whose combination does not have this property? A mathematical world can be extended to a bigger one. Could it be that objects with a certain property from the basic world, loose this property with respect to the bigger world? There are many infinities, and properties related to the smallest one, have been already defined. How to go to higher infinities? The project deals with objects that are subsets of the real line, and combinatorial covering properties, i.e., some properties following from different disciplines of mathematics. It is now one of the most active streams of research within pure mathematics and their foundations. Our goals are to answer the above questions whose professional reformulations are central problems of the field. Some methods that are planned to be used to attack the above problems are very far from being exhausted, and we want to use them in a comprehensive manner. For instance, forcing is a method for proving undecidability of statements, and it extents mathematical worlds to bigger ones. The powerful tools of this modern topic have not been applied widely to the considered field. Any new technique of constructions of sets with considered properties would be a remarkable input into the theory - thus far, a very few methods are available. The final problem about higher infinities, is completely new. It should be checked what are possible generalizations of classic properties, and what is the relation between this higher context and the classic one. Expected output of the project will develop the combinatorial covering properties theory in itself. Since this theory connects various mathematical branches and makes it possible to transport and apply methods from each of these fields to the other ones, the influence and importance of achieved goals can be even greater than suggested in the project description.

The project investigated which properties of sets of real numbers are determined by the standard axioms of mathematics and which depend on the particular mathematical universe in which they are studied. Questions of this kind lie at the heart of modern set theory and reflect the fact that some mathematical statements can neither be proved nor disproved from the usual axioms alone. The project focused on combinatorial covering properties of sets of real numbers. Roughly speaking, these properties describe how complicated sets can be covered by simple open sets and are central to the interplay between topology and set theory. The main objective of the project was to develop a systematic understanding of classical combinatorial covering properties in several of the most important mathematical universes. Rather than establishing isolated independence results, the aim was to obtain a coherent picture of the relationships between the different covering properties and of the ways in which they change under different mathematical assumptions. One major outcome concerns the Sacks model. We showed that sets with the Menger property satisfy a version of the perfect set property. Informally, this means that such sets either possess a comparatively regular structure or must necessarily be very large. In addition, we obtained a complete description of the Hurewicz property and several closely related combinatorial covering properties in this mathematical universe. Another major outcome concerns the Miller model. Here we established new structural results for several classes of "small" sets and answered several open questions. In particular, we discovered new connections between the Hurewicz property, the Rothberger property, and other classical combinatorial covering properties. Furthermore, our work on concentrated sets led to a better understanding of the interplay between these properties and of their dependence on the underlying mathematical axioms. The results answer several open questions concerning classical combinatorial covering properties, reveal new connections between topology and set theory, and provide new tools for future research on related problems. Overall, they contribute to a substantially clearer understanding of the behaviour of classical combinatorial covering properties in different mathematical universes.