arXiv subjects
Publications and source records attributed to Ol'ga V. Sipacheva.
It is proved that the free Boolean topological group $B(X)$ on a Tychonoff space $X$ is Weil complete if and only if the space $X$ is Dieudonné complete. This result provides a positive answer to a question posed by the first listed author in 2015.
A brief introduction to universal algebra and the theory of topological algebras, their varieties, and free topological algebras is presented. Free topological Mal'tsev algebras are studied. Their properties, relationship with topological groups and heaps, and the problem of direct limit decomposition are considered.
Right and left thick, syndetic, piecewise syndetic, and fat sets in groups are studied. The main concern is the interplay between such sets in Boolean groups. Natural topologies closely related to fat sets are also considered, which leads to interesting relations between fat sets and ultrafilters.
Families of unconditionally $τ$-closed and $τ$-algebraic sets in a group are defined, which are natural generalizations of unconditionally closed and algebraic sets defined by Markov. A sufficient condition for the coincidence of these families is found. In particular, it is proved that these families coincide in any group of cardinality at most $τ$. This result generalizes both Markov's theorem on the coincidence of unconditionally closed and algebraic sets in a countable group (as is known, they may be different in an uncountable group) and Podewski's theorem on the topologizablity of any ungebunden group.
It is proved that the continuum hypothesis implies the existence of a group M containing a nonalgebraic unconditionally closed set, i.e., a set which is closed in any Hausdorff group topology on M but is not an intersection of finite unions of solution sets of equations in M.
It is proved that, in certain subgroups of direct products of countable groups, the property of being an unconditionally closed set coincides with that of being an algebraic set. In particular, these properties coincide in all Abelian groups.
An embeddability criterion for zero-dimensional metrizable topological spaces in zero-dimensional metrizable topological groups is given. A space which can be embedded as a closed subspace in a zero-dimensional metrizable group but is not strongly zero-dimensional is constructed; thereby, an example of a metrizable group with noncoinciding dimensions ind and dim is obtained. It is proved that one of Kulesza's zero-dimensional metrizable spaces cannot be embedded in a metrizable zero-dimensional group.