Ultrafiltros y el problema de grupos topológicos extremadamente disconexos (2020)
In 1967 Arhangel skii wondered if there are nondiscrete extremally disconnected topological groups. Soon after, Sirota showed that, for the countable case, its existence is consistent by assuming CH. Several of the results found later showed a relationship between these groups and ultrafilters with some combinatorial property. In 2017 the problem was solved by Reznichenko and Sipacheva for the countable case, by showing that the existence of one of these groups implies the existence of a rapid filter. In the work some of the main results that relate countable nondiscrete extremally disconnected topological groups with ultrafilters with combinatorial properties are explored. To begin, the filters that will be of interest to us are exposed, various definitions are given for each one and how they are related one to another. After this, we study some left invariant topologies with a certain maximality property, built from a given filter; then we use these topologies to show the existence of extremally disconnected left topological groups. We continue with the study of discrete sets in countable topological groups, where we show a series of results on discrete sequences and then we strengthen some of these by restricting to the case of Boolean groups. In the last chapter we demonstrate the results that directly relate the filters to the nondiscrete extremally disconnected topological groups. We begin by showing that if there is a selective ultrafilter, then it is possible to construct an extremally disconnected topological group. Then, we proceed to show that under some extra hypotheses, the existence of a countable nondiscrete extremally disconnected topological group implies the existence of a P-point or a selective ultrafilter. We conclude the work with the theorem that states that the existence of such groups implies the existence of a rapid filter.