arXiv subjects

Alexander V. Osipov

Publications and source records attributed to Alexander V. Osipov.

At least 19 recordsLinked to original sources

arXiv2026

Some function applications of weak $λ$-spaces

In this paper it is proved that there exists (in $ZFC$) a separable pseudocompact weak $λ$-space $X$ which is not $Δ_1$-space. It follows that there exists a space $X$ such that $B_1(X,[0, 1])$ is Choquet, while $B_1(X)$ is not Choquet. Also it is proved that there exists a $γ$-set $X$ which is not weak $λ$-set. It follows that there exists a zero-dimensional separable metrizable space $X$ such that $B_1(X)$ is Baire, while $B_1(X, [0,1])$ is not Choquet. These results answers the previously posed questions.

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arXiv2026

Baire-type properties of topological vector spaces

Burzyk, Kliś and Lipecki proved that every topological vector space (tvs) $E$ with the property $(K)$ is a Baire space. Kcakol and Sánchez Ruiz proved that every sequentially complete Fréchet--Urysohn locally convex space (lcs) is Baire. Being motivated by the property $(K)$ and the notion of a Mackey null sequence we introduce a property $(MK)$ which is strictly weaker than the property $(K)$, and show that any locally complete lcs has the property $(MK)$. We prove that any $κ$-Fréchet--Urysohn tvs with the property $(MK)$ is a Baire space; consequently, each locally complete $κ$-Fréchet--Urysohn lcs is a Baire space. This generalizes both the aforementioned results. We construct a feral Baire space $E$ with the property $(K)$ and which is not $κ$-Fréchet--Urysohn. Although a $κ$-Fréchet--Urysohn lcs $E$ can be not a Baire space, we show that $E$ is always $b$-Baire-like in the sense of Ruess. Applications to spaces of Baire functions and $C_k$-spaces are given.

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arXiv2026

On $κ$-Frechet-Urysohn topological groups

We characterize $κ$-Fréchet--Urysohn topological groups. Using this characterization we show that: (1) a hemicompact topological group is $κ$-Fréchet--Urysohn iff it is locally compact, and (2) if $F$ is a closed metrizable subspace of a topological vector space (tvs) $E$ such that the quotient $E/F$ is a $κ$-Fréchet--Urysohn space, then also $E$ is a $κ$-Fréchet--Urysohn space. Consequently, the product of a $κ$-Fréchet--Urysohn tvs and a metrizable tvs is a $κ$-Fréchet--Urysohn space. Under Martin's Axiom, we construct a countable Boolean $κ$-Fréchet--Urysohn group which is not a $k_{\mathbb R}$-space.

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arXiv2026

Completeness and reflexivity type properties of $B_1(X)$

For a Tychonoff space $X$, $B_1(X)$ denotes the space of all Baire-one functions on $X$ endowed with the pointwise topology. We prove that the following assertions are equivalent: (1) $B_1(X)$ is a (semi-)Montel space, (2) $B_1(X)$ is a (semi-)reflexive space, (3) $B_1(X)$ is a (quasi-)complete space, (4) $B_1(X)=\mathbb{R}^X$, (5) $X$ is a $Q_f$-space. It is proved that $B_1(X)$ is sequentially complete iff $B_1(X)$ is locally complete iff $X$ is a $CZ$-space. In the case when $K$ is a compact space, we show that $B_1(K)$ is locally complete iff $K$ is scattered. We thoroughly study the case when $X$ is a separable metrizable space. Numerous distinguished examples are given.

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arXiv2025

Various $S(n)$-closednesses in $S(n)$-spaces with examples

In this paper we continue to study various types of closures in $S(n)$-spaces. The main results are related to the construction and illustration of examples that allow us to understand the relationship between $S(n)$-closed, $S(n)$-$θ$-closed, weakly $S(n)$-closed and weakly $S(n)$-$θ$-closed spaces for each $n\in \mathbb{N}$. The relation of these classes in Lindelöf spaces is shown. Some of the solved problems formulated by D. Dikranjan and E. Giuli are presented in the examples.

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arXiv2025

The $κ$-Fréchet-Urysohn property for $C_p(X)$ is equivalent to Baireness of $B_1(X)$

A topological space $X$ is Baire if the intersection of any sequence of open dense subsets of $X$ is dense in $X$. We establish that the property $(κ)$ for a Tychonoff space $X$ is equivalent to Baireness of $B_1(X)$ and, hence, the Banakh property for $C_p(X)$ is equivalent to meagerness of $B_1(X)$. Thus, we obtain one characteristic of the Banakh property for $C_p(X)$ through the property of space $X$.

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arXiv2025

The $Δ_1$-property of $X$ is equivalent to the Choquet property of $B_1(X)$

We give a characterization of the $Δ_1$-property of any Tychonoff space $X$ in terms of the function space $B_1(X)$ of all Baire-one real-valued functions on a space $X$ with the topology of pointwise convergence. We establish that for a Tychonoff space $X$ the $Δ_1$-property is equivalent to the Choquet property of $B_1(X)$. Also we construct under $ZFC$ an example of a separable pseudocompact space $X$ such that $C_p(X)$ is $κ$-Frechet-Urysohn but $X$ fails to be a $Δ_1$-space. This answers a question of Kakol-Leiderman-Tkachuk.

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arXiv2025

On Baire property, compactness and completeness properties of spaces of Baire functions

A topological space $X$ is Baire if the intersection of any sequence of open dense subsets of $X$ is dense in $X$. One of the interesting problems for the space of Baire functions is the Banakh-Gabriyelyan problem: Let $α$ be a countable ordinal. Characterize topological spaces $X$ and $Y$ for which the function space $B_α(X,Y)$ is Baire. In this paper, for any Frechet space $Y$ , we have obtained a characterization topological spaces $X$ for which the function space $B_α(X,Y)$ is Baire. In particular, we proved that $B_α(X,\mathbb{R})$ is Baire if and only if $B_α(X,Y)$ is Baire for any Banach space $Y$. Also we proved that many completeness and compactness properties coincide in spaces $B_α(X,Y)$ for any Frechet space $Y$.

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arXiv2025

Tightness type properties of spaces of quasicontinuous functions

Using approximation by continuous functions we prove the following statements to types of tightness in a space $Q_p(X, \mathbb{R})$ of all quasicontinuous real-valued functions with the topology $τ_p$ of pointwise convergence: the countability of tightness (fan-tightness, strong fan-tightness) at a point $f$ of space $Q_p(X, \mathbb{R})$ implies the countability of tightness (fan-tightness, strong fan-tightness) of space $Q_p(X,Y)$ of all quasicontinuous functions from $X$ into any non-one-point metrizable space $Y$. This result is the answer to the open question in the class of metrizable spaces.

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arXiv2024

Baire property of space of Baire-one functions

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 $X$. One of the interesting problems for the space $B_1(X)$ of all Baire-one real-valued functions is characterization topological space $X$ for which the function space $B_1(X)$ is Baire. In this paper, we solve this problem, namely, we have obtained a characterization when a function space $B_1(X)$ has the Baire property for any Tychonoff space $X$. Also we proved that $B_1(X)$ is Baire for any $γ$-space $X$. This answers a question posed recently by T. Banakh and S. Gabriyelyan. We also conclude that, it is consistent there are no uncountable separable metrizable space $X$ such that $B_1(X)$ is countable dense homogeneous.

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arXiv2024

On Baire property of spaces of compact-valued measurable functions

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 $X$. One of the interesting problems in the theory of functional spaces is the characterization of the Baire property of a functional space through the topological property of the support of functions. In the paper this problem is solved for the space $M(X, K)$ of all measurable compact-valued ($K$-valued) functions defined on a measurable space $(X,Σ)$ with the topology of pointwise convergence. It is proved that $M(X, K)$ is Baire for any metrizable compact space $K$.

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arXiv2024

Dieudonné completeness of function spaces

A space is called Dieudonné complete if it is complete relative to the maximal uniform structure compatible with its topology. In this paper, we investigated when 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 $X$ is Dieudonné complete. Also we proved a generalization of the Eberlein-Šmulian theorem to the class of Banach spaces.

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arXiv2024

Velichko's notions close to sequentially separability and their hereditary variants in $C_p$-theory

A space $X$ is sequentially separable if there is a countable $S\subset X$ such that every point of $X$ is the limit of a sequence of points from $S$. In 2004, N.V. Velichko defined and investigated concepts close to sequentially separability: $σ$-separability and $F$-separability. The aim of this paper is to study $σ$-separability and $F$-separability (and their hereditary variants) of the space $C_p(X)$ of all real-valued continuous functions, defined on a Tychonoff space $X$, endowed with the pointwise convergence topology. In particular, we proved that $σ$-separability coincides with sequential separability. Hereditary variants (hereditarily $σ$-separablity and hereditarily $F$-separablity) coincides with Frechet-Urysohn property in the class of cosmic spaces.

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arXiv2024

On the first Banach problem, concerning condensations of absolute $κ$-Borel sets onto compacta

It is consistent that the continuum be arbitrary large and no absolute $κ$-Borel set $X$ of density $κ$, $\aleph_1<κ<\mathfrak{c}$, condenses onto a compact metric space. It is consistent that the continuum be arbitrary large and any absolute $κ$-Borel set $X$ of density $κ$, $κ\leq\mathfrak{c}$, containing a closed subspace of the Baire space of weight $κ$, condenses onto a compactum. In particular, applying Brian's results in model theory, we get the following unexpected result. Given any $A\subseteq \mathbb{N}$ with $1\in A$, there is a forcing extension in which every absolute $\aleph_n$-Borel set, containing a closed subspace of the Baire space of weight $\aleph_n$, condenses onto a compactum if, and only if, $n\in A$.

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arXiv2024

On countable tightness type properties of spaces of quasicontinuous functions

In this paper we get characterizations countable tightness, countable fan-tightness and countable strong fan-tightness of spaces of quasicontinuous functions with the topology of pointwise convergence from a open Whyburn $T_2$-space $X$ into the discrete two-point space $\{0, 1\}$ through properties of $X$ determined by selection principles. These properties (e.g. $S_1(K, K)$, $K_Ω$-Lindelofness, $S_1(K_Ω, K_Ω)$) were defined by M. Scheepers and studied in theory of selection principles in the class of metric spaces. For any cardinal number $κ$, we get a functional characterization of $κ^+$-Lusin space in class of separable metrizable spaces through tightness of compact subsets of a space of quasicontinuous real-valued functions with the topology of pointwise convergence.

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arXiv2023

On the product of almost discrete Grothendieck spaces

A topological space $X$ is called almost discrete, if it has precisely one nonisolated point. In this paper, we get that for a countable product $X=\prod X_i$ of almost discrete spaces $X_i$ the space $C_p(X)$ of continuous real-valued functions with the topology of pointwise convergence is a $μ$-space if, and only if, $X$ is a weak $q$-space if, and only if, $t(X)=ω$ if, and only if, $X$ is functionally generated by the family of all its countable subspaces. This result makes it possible to solve Archangel'skii's problem on the product of Grothendieck spaces. It is proved that in the model of $ZFC$, obtained by adding one Cohen real, there are Grothendieck spaces $X$ and $Y$ such that $X\times Y$ is not weakly Grothendieck space. In $(PFA)$: the product of any countable family almost discrete Grothendieck spaces is a Grothendieck space.

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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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