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Mikhail Tkachenko

Publications and source records attributed to Mikhail Tkachenko.

8 recordsLinked to original sources
arXiv2025

Regular rigid Korovin orbits

An example of an infinite regular feebly compact quasitopological group is presented such that all continuous real-valued functions on the group are constant. The example is based on the use of Korovin orbits in $X^G$, where $X$ is a special regular countably compact space constructed by S.Bardyla and L.Zdomskyy and $G$ is an abstract Abelian group of an appropriate cardinality. Also, we study the interplay between the separation properties of the space $X$ and Korovin orbits in $X^G$. We show in particular that if $X$ contains two nonempty disjoint open subsets, then every Korovin orbit in $X^G$ is Hausdorff.

math.GN↗
arXiv2020

Weak completions of paratopological groups

Given a $T_0$ paratopological group $G$ and a class $\mathcal C$ of continuous homomorphisms of paratopological groups, we define the $\mathcal C$-$semicompletion$ $\mathcal C[G)$ and $\mathcal C$-$completion$ $\mathcal C[G]$ of the group $G$ that contain $G$ as a dense subgroup, satisfy the $T_0$-separation axiom and have certain universality properties. For special classes $\mathcal C$, we present some necessary and sufficient conditions on $G$ in order that the (semi)completions $\mathcal C[G)$ and $\mathcal C[G]$ be Hausdorff. Also, we give an example of a Hausdorff paratopological abelian group $G$ whose $\mathcal C$-semicompletion $\mathcal C[G)$ fails to be a $T_1$-space, where $\mathcal C$ is the class of continuous homomorphisms of sequentially compact topological groups to paratopological groups. In particular, the group $G$ contains an $\omega$-bounded sequentially compact subgroup $H$ such that $H$ is a topological group but its closure in $G$ fails to be a subgroup.

math.GR↗
arXiv2019

Simply $sm$-factorizable (para)topological groups and their completions

Let us call a (para)topological group \emph{strongly submetrizable} if it admits a coarser separable metrizable (para)topological group topology. We present a characterization of simply $sm$-factorizable (para)topo\-logical groups by means of continuous real-valued functions. We show that a (para)topo\-logical group $G$ is a simply $sm$-factorizable if and only if for each continuous function $f\colon G\to \mathbb{R}$, one can find a continuous homomorphism $\varphi$ of $G$ onto a strongly submetrizable (para)topological group $H$ and a continuous function $g\colon H\to \mathbb{R}$ such that $f=g\circ\varphi$. This characterization is applied for the study of completions of simply $sm$-factorizable topological groups. We prove that the equalities $\mu{G}=\varrho_\omega{G}=\upsilon{G}$ hold for each Hausdorff simply $sm$-factorizable topological group $G$. This result gives a positive answer to a question posed by Arhangel'skii and Tkachenko in 2018. Also, we consider realcompactifications of simply $sm$-factorizable paratopological groups. It is proved, among other results, that the realcompactification, $\upsilon{G}$, and the Dieudonn\'e completion, $\mu{G}$, of a regular simply $sm$-factorizable paratopological group $G$ coincide and that $\upsilon{G}$ admits the natural structure of paratopological group containing $G$ as a dense subgroup and, furthermore, $\upsilon{G}$ is also simply $sm$-factorizable. Some results in [\emph{Completions of paratopological groups, Monatsh. Math. \textbf{183} (2017), 699--721}] are improved or generalized.

math.GN↗
arXiv2012

Reflexivity in precompact groups and extensions

We establish some general principles and find some counter-examples concerning the Pontryagin reflexivity of precompact groups and P-groups. We prove in particular that: (1) A precompact Abelian group G of bounded order is reflexive iff the dual group $\hat{G}$ has no infinite compact subsets and every compact subset of G is contained in a compact subgroup of G. (2) Any extension of a reflexive P-group by another reflexive P-group is again reflexive. We show on the other hand that an extension of a compact group by a reflexive $ω$-bounded group (even dual to a reflexive P-group) can fail to be reflexive. We also show that the P-modification of a reflexive $σ$-compact group can be nonreflexive (even if the P-modification of a locally compact Abelian group is always reflexive).

math.GN↗
arXiv2011

Pontryagin duality in the class of precompact Abelian groups and the Baire property

We present a wide class of reflexive, precompact, non-compact, Abelian topological groups $G$ determined by three requirements. They must have the Baire property, satisfy the \textit{open refinement condition}, and contain no infinite compact subsets. This combination of properties guarantees that all compact subsets of the dual group $G^\wedge$ are finite. We also show that many (non-reflexive) precompact Abelian groups are quotients of reflexive precompact Abelian groups. This includes all precompact almost metrizable groups with the Baire property and their products. Finally, given a compact Abelian group $G$ of weight $\geq 2^\om$, we find proper dense subgroups $H_1$ and $H_2$ of $G$ such that $H_1$ is reflexive and pseudocompact, while $H_2$ is non-reflexive and almost metrizable.

math.GN↗
arXiv2010

Nondiscrete P-Groups Can be Reflexive

We present a series of examples of nondiscrete reflexive P-groups (i.e., groups in which all $G_δ$-sets are open) as well as noncompact reflexive $ω$-bounded groups (in which the closure of every countable set is compact). Our main result implies that every product of feathered (equivalently, almost metrizable) Abelian groups equipped with the P-modified topology is a reflexive group. In particular, every compact Abelian group with the P-modified topology is reflexive. This answers a question posed by S. Hernández and P. Nickolas and solves a problem raised by Ardanza-Trevijano, Chasco, Dom\'ınguez, and Tkachenko.

math.GN↗