Spectra of combinatorial covering properties

This project concentrates on properties of subsets of the real line which have both combinatorial and topological nature, i.e., are preserved when we apply some continuous transformations. One of the ways to study a topological property is to investigate the sizes or, mathematically speaking, cardinalities of the objects possessing it. This line of research may be traced back to by now almost 100 years old investigations of the upper bounds of cardinalities of topological spaces based on their other quantitative characteristics usually describing both the local and global complexity of their topology. Another theme, which is more typical for properties of sets of reals, is investigating the minimal cardinalities of the corresponding examples. Later the combination of these two approaches was gaining more popularity, leading to studies of spectra of topological properties, namely the collection of all cardinalities for which there is a representative having the property in question. There are two approaches here: On the one hand side, on can try to construct concrete examples of objects with the needed cardinalities. However, we also plan to use existential arguments in order to show that the spectrum of some property must be rich in an appropriate sense. Of course the aforementioned studies cannot be pursued solely on the basis of the standard axioms of mathematics formulated by Zermello and Fraenkel since Gödel proved that these do not exclude the possibility that there are no intermediate cardinalities of sets of reals between those of natural numbers and the whole real line. Thus, we intend to work in specially designed mathematical universes, where there are many mutually different infinities between the two mentioned above. The crucial way to extend a mathematical world to a bigger one uses forcing, the method invented by Cohen 60 years ago. This powerful tool has not been used widely in our context, but it found striking applications in closely related areas which allows us to hope for similarly broad effects also in the cases we plan to consider. Besides spectra, we hope that the expected output of the project will further develop the selection principles theory. Since this theory connects several mathematical branches and enables to translate and use methods from each of these to the other ones, the actual achievements in the framework of the project can be even greater than described in the project description.