arXiv subjects

Alexander V. Osipov

Publications and source records attributed to Alexander V. Osipov.

At least 37 records · Page 2Linked to original sources

arXiv2023

Joint continuity in semitopological monoids and semilattices

In this paper we study the separately continuous actions of semitopological monoids on pseudocompact spaces. The main aim of this paper is to generalize Lawson's results to some class of pseudocompact spaces. Also, we introduce a concept of a weak $q_D$-space and prove that a pseudocompact space and a weak $q_D$-space form a Grothendieck pair. As an application of the main result, we investigate the continuity of multiplication and taking inverses in subgroups of semitopological semigroups. In particular, we get that if $(S,\bullet)$ is a Tychonoff pseudocompact semitopological monoid with a quasicontinuous multiplication $\bullet$ and $G$ is a subgroup of $S$, then $G$ is a topological group. Also, we study the continuity of operations in semitopological semilattices.

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arXiv2023

Generalization of the Grothendieck's theorem

In this paper, we have obtained a generalization of the Grothendieck's theorem for the space of continuous mappings $C_{λ,μ}(X,Y)$ where $Y$ is a complete uniform space with the uniformity $μ$ endowed with the topology of uniform convergence on the family $λ$ of subsets of $X$. A new topological game is defined - the Asanov-Velichko game, which makes it possible to single out a class of topological spaces of the Grothendieck type. The developed technique is used to generalize the Grothendieck theorem for the space of continuous mappings endowed with the set-open topology.

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arXiv2023

The projectively Hurewicz property is t-invariant

A space $X$ is projectively Hurewicz provided every separable metrizable continuous image of $X$ is Hurewicz. In this paper we prove that the projectively Hurewicz property is $t$-invariant, i.e., if $C_p(X)$ is homeomorphic to $C_p(Y)$ and $X$ is projectively Hurewicz, then $Y$ is projectively Hurewicz, too.

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arXiv2023

Embedding Theorems for function spaces

In this paper, we have proved results similar to Tychonoff's Theorem on embedding a space of functions with the topology of pointwise convergence into the Tychonoff product of topological spaces, but applied to the function space $C(X,Y)$ of all continuous functions from a topological space $X$ into a uniform space $Y$ with the topology of uniform convergence on a family of subsets of $X$ and with the (weak) set-open topology. We also investigated the following question: how the topological embedding of the space $C(X,Y)$ is related to algebraic structures (such as topological groups, topological rings and topological vector spaces) on $C(X,Y)$.

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arXiv2023

Some observations on a clopen version of the Rothberger property

In this paper, we proved that a clopen version $S_1(C_O, C_O)$ of the Rothberger property and Borel strong measure zeroness are independent. For a zero-dimensional metric space $(X, d)$, $X$ satisfies $S_1(C_O, C_O)$ if, and only if, $X$ has Borel strong measure zero with respect to each metric which has a same topology as d has. In a zero-dimensional space, the game $G_1(O,O)$ is equivalent to the game $G_1(C_O, C_O)$ and the point-open game is equivalent to the point-clopen game. Using reflections, we obtained that the game $G_1(C_O, C_O)$ and the point-clopen game are strategically and Markov dual. An example is given for a space on which the game $G_1(C_O, C_O)$ is undetermined.

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arXiv2023

Covering games using semi-open sets

In this paper, we prove the following Theorems 1. An extremally disconnected space $X$ has the semi-Menger property if and only if One does not have a winning strategy in the game $G_{fin}(sO,sO)$. 2. An extremally disconnected space $X$ has the semi-Rothberger property if and only if One does not have a winning strategy in the game $G_1(sO,sO)$.

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arXiv2023

Frechet-Urysohn property of quasicontinuous functions

The aim of this paper is to study the Frechet-Urysohn property of the space $Q_p(X,\mathbb{R})$ of real-valued quasicontinuous functions, defined on a Hausdorff space $X$, endowed with the pointwise convergence topology. It is proved that under Suslin's Hypothesis, for an open Whyburn space $X$, the space $Q_p(X,\mathbb{R})$ is Frechet-Urysohn if and only if $X$ is countable. In particular, it is true in the class of first-countable regular spaces $X$. In ZFC, it is proved that for a metrizable space $X$, the space $Q_p(X,\mathbb{R})$ is Frechet-Urysohn if and only if $X$ is countable.

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arXiv2022

Note on the Banach Problem 1 of condensations of Banach spaces onto compacta

It is consistent with any possible value of the continuum $\mathfrak{c}$ that every infinite-dimensional Banach space of density $\leq \mathfrak{c}$ condenses onto the Hilbert cube. Let $μ$ be a cardinal of uncountable cofinality. It is consistent that the continuum be arbitrary large, no Banach space $X$ of density $γ$, $μ< γ< \mathfrak{c}$ condenses onto a compactum, but any Banach space of density $μ$ admit a condensation onto a compactum. In particular, for $μ=ω_1$, it is consistent that $\mathfrak{c}$ is arbitrarily large, no Banach space of density $γ$, $ω_1< γ< \mathfrak{c}$, condenses onto a compactum. These results imply a complete answer to the Problem 1 in the Scottish Book for Banach spaces: When does a Banach space X admit a bijective continuous mapping onto a compact metric space?

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arXiv2022

Baire property of some function spaces

A compact space $X$ is called $π$-monolithic if for any surjective continuous mapping $f:X\rightarrow K$ where $K$ is a metrizable compact space there exists a metrizable compact space $T\subseteq X$ such that $f(T)=K$. A topological space $X$ is Baire if the intersection of any sequence of open dense subsets of $X$ is dense in $X$. Let $C_p(X,Y)$ denote the space of all continuous $Y$- valued functions $C(X,Y)$ on a Tychonoff space $X$ with the topology of pointwise convergence. In this paper we have proved that for a totally disconnected space $X$ the space $C_p(X,\{0,1\})$ is Baire if, and only if, $C_p(X,K)$ is Baire for every $π$-monolithic compact space $K$. For a Tychonoff space $X$ the space $C_p(X)$ is Baire if, and only if, $C_p(X,L)$ is Baire for each Frechet space $L$. We construct a totally disconnected Tychonoff space $T$ such that $C_p(T,M)$ is Baire for a separable metric space $M$ if, and only if, $M$ is a Peano continuum. Moreover, $C_p(T,[0,1])$ is Baire but $Cp(T,\{0,1\})$ is not.

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arXiv2022

Baireness of the space of pointwise stabilizing functions of the first Baire class

A topological space $X$ is Baire if the Baire Category Theorem holds for $X$, i.e., the intersection of any sequence of open dense subsets of $X$ is dense in $X$. In this paper, we have obtained that the space $B^{st}_1(X)$ of pointwise stabilizing Baire-one functions is Baire if the space $B_1(X)$ of Baire-one functions is so. This answers a question posed recently by T. Banakh and S. Gabriyelyan.

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arXiv2022

Spaces of Real Functions, Covers and Dense Subsets

In paper we study relationships between covering properties of a topological space $X$ and the space $(USC^*(X),τ_{\mathcal{B}})$ of bounded upper semicontinuous functions on $X$ with the topology $τ_{\mathcal{B}}$ defined by the bornology $\mathcal{B}$ on $X$. We present characterizations of various local properties of $(USC^*(X),τ_{\mathcal{B}})$ in terms of selection principles related to bornological covers of $X$. We also investigate topological properties of sequences of upper dense and upper sequentially dense subsets of $(USC^*(X),τ_{\mathcal{B}})$.

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arXiv2022

Baire property of spaces of $[0,1]$-valued continuous functions

A topological space $X$ is Baire if the intersection of any sequence of open dense subsets of $X$ is dense in $X$. Let $C_p(X,[0,1])$ denote the space of all continuous $[0,1]$-valued functions on a Tychonoff space $X$ with the topology of pointwise convergence. In this paper, we have obtained a characterization when the function space $C_p(X,[0,1])$ is Baire for a Tychonoff space $X$ all separable closed subsets of which are $C$-embedded. In particular, this characterization is true for normal spaces and, hence, for metrizable spaces. Moreover, we obtained that the space $C_p(X,[0,1])$ is Baire, if and only if, the space $Cp(X,K)$ is Baire for a Peano continuum $K$.

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arXiv2022

Mildly version of Hurewicz Basis covering property and Hurewicz measure zero spaces

In this paper, we introduced the mildly version of the Hurewicz basis covering property, studied by Babinkostova, Kočinac, and Scheepers. A space $X$ is said to have mildly-Hurewicz property if for each sequence $\langle \mathcal{U}_n : n\in ω\rangle$ of clopen covers of $X$ there is a sequence $\langle \mathcal{V}_n : n\in ω\rangle$ such that for each $n$, $\mathcal{V}_n$ is a finite subset of $\mathcal{U}_n$ and for each $x\in X$, $x$ belongs to $\bigcup\mathcal{V}_n$ for all but finitely many $n$. Then we characterized mildly-Hurewicz property by mildly-Hurewicz Basis property and mildly-Hurewicz measure zero property for metrizable spaces.

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arXiv2022

Every metric space of weight $λ=λ^{\aleph_0}$ admits a condensation onto a Banach space

In this paper, we have proved that for each cardinal number $λ$ such that $λ=λ^{\aleph_0}$ a metric space of weight $λ$ admits a bijective continuous mapping onto a Banach space of weight $λ$. Then, we get that every metric space of weight continuum admits a bijective continuous mapping onto the Hilbert cube. This resolves the famous Banach's Problem (when does a metric (possibly Banach) space $X$ admit a bijective continuous mapping onto a compact metric space?) in the class of metric spaces of weight continuum. Also we get that every metric space of weight $λ=λ^{\aleph_0}$ admits a bijective continuous mapping onto a Hausdorff compact space. This resolves the Alexandroff Problem (when does a Hausdorff space $X$ admit a bijective continuous mapping onto a Hausdorff compact space?) in the class of metric spaces of weight $λ=λ^{\aleph_0}$.

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arXiv2022

Some observations on the mildly Menger property and topological games

In this paper, we defined two new games - the mildly Menger game and the compact-clopen game. In a zero-dimensional space, the Menger game is equivalent to the mildly Menger game and the compact-open game is equivalent to the compact-clopen game. An example is given for a space on which the mildly Menger game is undetermined. Also we introduced a new game namely K-quasi-component-clopen game and proved that this game is equivalent to the compact-clopen game. Then we proved that if a topological space is a union of countably many quasi-components of compact sets, then TWO has a winning strategy in the mildly Menger game.

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arXiv2021

Various types of completeness in topologized semilattices

A topologized semilattice $X$ is called complete if each non-empty chain $C\subset X$ has $\inf C$ and $\sup C$ that belong to the closure $C$ of the chain $C$ in $X$. In this paper, we introduce various concepts of completeness of topologized semilattices in the context of operators that generalize the closure operator, and study their basic properties. In addition, examples of specific topologized semilattices are given, showing that these classes do not coincide with each other. Also in this paper, we prove theorems that allow us to generalize the available results on complete semilattices endowed with topology.

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arXiv2020

Compact condensations of Hausdorff spaces

In this paper, we continue to study one of the classic problems in general topology raised by P.S. Alexandrov: when a Hausdorff space $X$ has a continuous bijection (a condensation) onto a compactum? We concentrate on the situation when not only $X$ but also $X\setminus Y$ can be condensed onto a compactum whenever the cardinality of $Y$ does not exceed certain $τ$.

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arXiv2020

Projective versions of the properties in the Scheepers Diagram

Let $\mathcal{P}$ be a topological property. A.V. Arhangel'skii calls $X$ projectively $\mathcal{P}$ if every second countable continuous image of $X$ is $\mathcal{P}$. Lj.D.R. Ko$\check{c}$inac characterized the classical covering properties of Menger, Rothberger, Hurewicz and Gerlits-Nagy in term of continuous images in $\mathbb{R}^ω$. In this paper we study the functional characterizations of all projective versions of the selection properties in the Scheepers Diagram.

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