ИСТИНА

The most important result of the report is the following recent theorem proved jointly with Evgenii Reznichenko (throughout, all topological groups are assumed to be Hausdorff). Theorem (Reznichenko and Sipacheva) (i) Any countable nondiscrete topological group whose identity element has nonrapid filter of neighborhoods contains a discrete subset with precisely one limit point. (ii) If there are no rapid filters, then any countable nondiscrete Boolean topological group contains two disjoint discrete subsets for each of which the zero of the group is a unique limit point. (A filter $\mathscr F$ on $\omega$ is rapid if every function $\omega\to \omega$ is majorized by the increasing enumeration of some element of $\mathscr F$. The nonexistence of rapid filters is consistent with ZFC.) This theorem and its consequences concerning countable topological groups with extremal properties answer, partially or completely, several old questions. In particular, they imply that the nonexistence of a nondiscrete separable extremally disconnected group, as well as that of a nondiscrete countable topological group in which all discrete subspaces are closed, is consistent with ZFC. A key role in the proof is played by fat subsets of a group. Fatness is a new notion of largeness organically related to that of syndeticity, which originated in Ramsey theory and topological dynamics in the context of the additive semigroup of positive integers, in which syndetic sets are precisely those with bounded gaps. In the study of countable topological groups with extremal properties, of most interest are Boolean groups (any extremally disconnected group and any Abelian or countable irresolvable group contains an open Boolean subgroup), and every Boolean group is free, being a linear space over the field $\mathbb Z_2$. Theorem. (i) Any countable Boolean topological group has a closed discrete basis. (ii) Any countable closed linearly independent set in an extremally disconnected Boolean group has at most one limit point. (iii) It is consistent with ZFC that any countable closed linearly independent set in an extremally disconnected Boolean group is discrete. Finally, we present a curious consequence of the equivalence between the existence of an extremally disconnected free (Boolean) topological group and that of selective ultrafilters, which concerns selectivity-type properties of filters.