No Arabic abstract
It is proved that any countable topological group in which the filter of neighborhoods of the identity element is not rapid contains a discrete set with precisely one nonisolated point. This gives a negative answer to Protasovs question on the existence in ZFC of a countable nondiscrete group in which all discrete subsets are closed. It is also proved that the existence of a countable nondiscrete extremally disconnected group implies the existence of a rapid ultrafilter and, hence, a countable nondiscrete extremally disconnected group cannot be constructed in ZFC.
It is proved that the existence of a countable extremally disconnected Boolean topological group containing a family of open subgroups whose intersection has empty interior implies the existence of a rapid ultrafilter.
Given an arbitrary measurable cardinal $kappa$, a nondiscrete Hausdorff extremally disconnected topological group of cardinality $kappa$ is constructed.
A group topology is said to be linear if open subgroups form a base of neighborhoods of the identity element. It is proved that the existence of a nondiscrete extremally disconnected group of Ulam nonmeasurable cardinality with linear topology implies that of a nondiscrete extremally disconnected group of cardinality at most $2^omega$ with linear topology.
Known and new results on free Boolean topological groups are collected. An account of properties which these groups share with free or free Abelian topological groups and properties specific of free Boolean groups is given. Special emphasis is placed on the application of set-theoretic methods to the study of Boolean topological groups.
We discuss various modifications of separability, precompactnmess and narrowness in topological groups and test those modifications in the permutation groups $S(X)$ and $S_{<omega}(X)$.