ИСТИНА

The main results of this talk stem from attempts to solve the following problem of Arhangel'skii. Problem (Arhangel'skii, 1967). Does there exist in ZFC a nondiscrete Hausdorff extremally disconnected topological group? Recall that a topological space is said to be extremally disconnected if the closure of any open set in this space is open (or, equivalently, the closures of any two disjoint open sets are disjoint). In the talk, a solution of Arhagel'skii's problem for countable groups is presented. Namely, it is proved that the nonexistence of a countable nondiscrete Hausdorff extremally disconnected group is consistent with ZFC. The proof of this assertion is based on the following two main theorems. Theorem 1. Any countable nondiscrete topological group whose identity element has nonrapid filter of neighborhoods contains a discrete sequence with precisely one limit point. Theorem 2. If there are no rapid filters, then any countable nondiscrete Hausdorff Boolean topological group contains two disjoint discrete subsets for each of which the zero of the group is a unique limit point. (A filter F on ω is said to be rapid if every function ω→ω is majorized by the increasing enumeration of some element of F. The nonexistence of rapid filters is consistent with ZFC.) These two theorems have a number of consequences concerning contable topological groups with extremal properties. For example, they imply that if there are no rapid ultrafilters, then any countable nondiscrete topological group is ω-resolvable (i.e., can be partitioned into countably many dense subsets) and that the nonexistence of countable nondiscrete nodec groups is consistent with ZFC (nodec means that all nowhere dense sets are closed).