arXiv subjects

Evgenii Reznichenko

Publications and source records attributed to Evgenii Reznichenko.

27 records · Page 2Linked to original sources

arXiv2023

Metrizability of CHART groups

For compact Hausdorff admissible right topological (CHART) group $G$, we prove $w(G)=πχ(G)$. This equality is well known for compact topological groups. This implies the criteria for the metrizability of CHART groups: if $G$ is first-countable (2013, Moors, Namioka) or $G$ is Fréchet (2013, Glasner, Megrelishvili), or $G$ has countable $π$-character (2022, Reznichenko) then $G$ is metrizable. Under the continuum hypothesis (CH) assumption, a sequentially compact CHART group is metrizable. Namioka's theorem that metrizable CHART groups are topological groups extends to CHART groups with small weight.

math.GR↗

arXiv2022

Algebraic structures on the Cantor set

Below, by space we mean a separable metrizable zero-dimensional space. It is studied when the space can be embedded in a Cantor set while maintaining the algebraic structure. Main results of the work: every space is an open retract of a Boolean precompact group; every strongly homogeneous space is rectifiable. In this case, the space can be embedded in the Cantor set with the preservation of the algebraic structure. An example of a strongly homogeneous space is constructed which do not admit the structure of a right topological group.

math.GN↗

arXiv2022

Extension of mappings from the product of pseudocompact spaces

Let $X$ and $Y$ be pseudocompact spaces and let the function $Φ: X\times Y\to \mathbb R$ be separately continuous. The following conditions are equivalent: (1) there is a dense $G_δ$ subset of $D\subset Y$ so that $Φ$ is continuous at every point of $X\times D$ (Namioka property); (2) $Φ$ is quasicontinuous; (3) $Φ$ extends to a separately continuous function on $βX\times βY$. This theorem makes it possible to combine studies of the Namioka property and generalizations of the Eberlein-Grothendieck theorem on the precompactness of subsets of function spaces. We also obtain a characterization of separately continuous functions on the product of several pseudocompact spaces extending to separately continuous functions on products of Stone-Cech extensions of spaces. These results are used to study groups and Mal'tsev spaces with separately continuous operations.

math.GN↗

arXiv2022

Continuity in right semitopological groups

Groups with a topology that is in consistent one way or another with the algebraic structure are considered. Classical groups with a topology are topological, paratopological, semitopological, and quasitopological groups. We also study other ways of matching topology and algebraic structure. The minimum requirement in this paper is that the group is a right semitopological group (such groups are often called right topological groups). We study when a group with a topology is a topological group; research in this direction began with the work of Deane Montgomery and Robert Ellis. (Invariant) semi-neighborhoods of the diagonal are used as a means of study.

math.GN↗

arXiv2022

Classes of Baire spaces defined by topological games

The article studies topological games that arise in the study of the continuity of operations in groups with topology, such as paratopological and semitopological groups. These games are modifications of the Banach--Mazur game. Given a two-player game $G(X)$ of the Banach--Mazur type, we define $Γ^G$-Baire, $Γ^G$-nonmeager and $Γ^G$-spaces. A space $X$ is a $Γ^G$-Baire if the second player does not have a winning strategy in $G(X)$. The classes of $Γ^G$-nonmeager spaces and $Γ^G$-spaces are defined similarly, with the help of modifications of the game $G(X)$. For the games under consideration, equivalent games are found, which facilitates studying the relationship between the resulting classes of spaces and determining which spaces belong to these classes. For this purpose, we introduce a modification of the Banach--Mazur game with four players. Results of this paper find application in the study the continuity of operations in groups with topology.

math.GN↗

arXiv2022

Continuity of the inverse

We define $Δ$-Baire spaces. If a paratopological group $G$ is $Δ$-Baire space, then $G$ is a topological group. Locally pseudocompact spaces, Baire $p$-spaces, Baire $Σ$-spaces, products of Čech-complete spaces are $Δ$-Baire spaces.

math.GN↗

arXiv2021

Functions on products of pseudocompact spaces

For every natural number $ n $, any continuous function on the product of $ X_1 \times X_2 \times ... \times X_n $ pseudocompact spaces extends to a separately continuous function on the product $ βX_1 \times βX_2 \times ... \times βX_n $ of their Stone-Cech compactifications.

math.GN↗

arXiv2018

Homogeneous Subspaces of Products of Extremally Disconnected Spaces

Homogeneous countably compact spaces $X$ and $Y$ whose product $X\times Y$ is not pseudocompact are constructed. It is proved that all compact subsets of homogeneous subspaces of the third power of an extremally disconnected space are finite. Moreover, under CH, all compact subsets of homogeneous subspaces of any finite power of an extremally disconnected space are finite and all compact subsets of homogeneous subspaces of the countable power of an extremally disconnected space are metrizable. It is also proved that all compact homogeneous subspaces of finite powers of an extremally disconnected space are finite, which strengthens Frol\'ık's theorem.

math.GN↗

arXiv2017

Discrete Subsets in Topological Groups and Countable Extremally Disconnected Groups

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 Protasov's 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.

math.GN↗