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Lyubomyr Zdomskyy

Publications and source records attributed to Lyubomyr Zdomskyy.

At least 37 records · Page 2Linked to original sources
arXiv2023

Countable spaces, realcompactness, and the pseudointersection number

All spaces are assumed to be Tychonoff. Given a realcompact space $X$, we denote by $\mathsf{Exp}(X)$ the smallest infinite cardinal $κ$ such that $X$ is homeomorphic to a closed subspace of $\mathbb{R}^κ$. Our main result shows that, given a cardinal $κ$, the following conditions are equivalent: $(1)$ There exists a countable crowded space $X$ such that $\mathsf{Exp}(X)=κ$, $(2)$ $\mathfrak{p}\leqκ\leq\mathfrak{c}$. In fact, in the case $\mathfrak{d}\leqκ\leq\mathfrak{c}$, every countable dense subspace of $2^κ$ provides such an example. This will follow from our analysis of the pseudocharacter of countable subsets of products of first-countable spaces. Finally, we show that a scattered space of weight $κ$ has pseudocharacter at most $κ$ in any compactification. This will allow us to calculate $\mathsf{Exp}(X)$ for an arbitrary (that is, not necessarily crowded) countable space.

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arXiv2023

On sequences of finitely supported measures related to the Josefson--Nissenzweig theorem

Given a Tychonoff space $X$, we call a sequence $\langleμ_n\colon n\inω\rangle$ of signed Borel measures on $X$ a finitely supported Josefson--Nissenzweig sequence (in short a JN-sequence) if: 1) for every $n\inω$ the measure $μ_n$ is a finite combination of one-point measures and $\|μ_n\|=1$, and 2) $\int_Xf\,\mathrm{d}μ_n\to0$ for every continuous function $f\in C(X)$. Our main result asserts that if a Tychonoff space $X$ admits a JN-sequence, then there exists a JN-sequence $\langleμ_n\colon n\inω\rangle$ such that: i) $\mbox{supp}(μ_n)\cap\mbox{supp}(μ_k)=\emptyset$ for every $n\neq k\inω$, and ii) the union $\bigcup_{n\inω}\mbox{supp}(μ_n)$ is a discrete subset of $X$. We also prove that if a Tychonoff space $X$ carries a JN-sequence, then either there is a JN-sequence $\langleμ_n\colon n\inω\rangle$ on $X$ such that $|\mbox{supp}(μ_n)|=2$ for every $n\inω$, or for every JN-sequence $\langleμ_n\colon n\inω\rangle$ on $X$ we have $\lim_{n\to\infty}|\mbox{supp}(μ_n)|=\infty$.

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arXiv2023

Grothendieck $C(K)$-spaces and the Josefson--Nissenzweig theorem

For a compact space $K$, the Banach space $C(K)$ is said to have the $\ell_1$-Grothendieck property if every weak* convergent sequence $\big\langleμ_n\colon\ n\inω\big\rangle$ of functionals on $C(K)$ such that $μ_n\in\ell_1(K)$ for every $n\inω$, is weakly convergent. Thus, the $\ell_1$-Grothendieck property is a weakening of the standard Grothendieck property for Banach spaces of continuous functions. We observe that $C(K)$ has the $\ell_1$-Grothendieck property if and only if there does not exist any sequence of functionals $\big\langleμ_n\colon\ n\inω\big\rangle$ on $C(K)$, with $μ_n\in\ell_1(K)$ for every $n\inω$, satisfying the conclusion of the classical Josefson--Nissenzweig theorem. We construct an example of a separable compact space $K$ such that $C(K)$ has the $\ell_1$-Grothendieck property but it does not have the Grothendieck property. We also show that for many classical consistent examples of Efimov spaces $K$ their Banach spaces $C(K)$ do not have the $\ell_1$-Grothendieck property.

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arXiv2023

Selective separability properties of Fréchet-Urysohn spaces and their products

In this paper we study the behaviour of selective separability properties in the class of Frechét-Urysohn spaces. We present two examples, the first one given in ZFC proves the existence of a countable Frechét-Urysohn (hence $R$-separable and selectively separable) space which is not $H$-separable; assuming $\mathfrak{p}=\mathfrak{c}$, we construct such an example which is also zero-dimensional and $α_{4}$. Also, motivated by a result of Barman and Dow stating that the product of two countable Frechét-Urysohn spaces is $M$-separable under PFA, we show that the MA is not sufficient here. In the last section we prove that in the Laver model, the product of any two $H$-separable spaces is $mH$-separable.

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arXiv2023

Ideal approach to convergence in functional spaces

We solve the last standing open problem from the seminal paper by J. Gerlits and Zs. Nagy, which was later reposed by A. Miller, T. Orenshtein and B. Tsaban. Namely, we show that under p = c there is a δ-set that is not a γ-set. Thus we construct a set of reals A such that the space Cp(A) of all real-valued continuous functions on A is not Frechet-Urysohn, but possesses the Pytkeev property. Moreover, under CH we construct a π-set that is not a δ-set solving a problem by M. Sakai. In fact, we construct various examples of δ-sets that are not γ-sets, satisfying finer properties parametrized by ideals on natural numbers. Finally, we distinguish ideal variants of the Frechet-Urysohn property for many different Borel ideals in the realm of functional spaces.

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arXiv2022

Convergence of measures after adding a real

We prove that if $\mathcal{A}$ is an infinite Boolean algebra in the ground model $V$ and $\mathbb{P}$ is a notion of forcing adding any of the following reals: a Cohen real, an unsplit real, or a random real, then, in any $\mathbb{P}$-generic extension $V[G]$, $\mathcal{A}$ has neither the Nikodym property nor the Grothendieck property. A similar result is also proved for a dominating real and the Nikodym property.

math.LO↗
arXiv2022

On complementability of $c_0$ in spaces $C(K\times L)$

Using elementary probabilistic methods, in particular a variant of the Weak Law of Large Numbers related to the Bernoulli distribution, we prove that for every infinite compact spaces $K$ and $L$ the product $K\times L$ admits a sequence $\langleμ_n\colon n\in\mathbb{N}\rangle$ of normalized signed measures with finite supports which converges to $0$ with respect to the weak* topology of the dual Banach space $C(K\times L)^*$. Our approach is completely constructive -- the measures $μ_n$ are defined by an explicit simple formula. We also show that this result generalizes the classical theorem of Cembranos and Freniche which states that for every infinite compact spaces $K$ and $L$ the Banach space $C(K\times L)$ contains a complemented copy of the space $c_0$.

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arXiv2022

On complemented copies of the space $c_0$ in spaces $C_p(X\times Y)$

Cembranos and Freniche proved that for every two infinite compact Hausdorff spaces $X$ and $Y$ the Banach space $C(X\times Y)$ of continuous real-valued functions on $X\times Y$ endowed with the supremum norm contains a complemented copy of the Banach space $c_{0}$. We extend this theorem to the class of $C_p$-spaces, that is, we prove that for all infinite Tychonoff spaces $X$ and $Y$ the space $C_{p}(X\times Y)$ of continuous functions on $X\times Y$ endowed with the pointwise topology contains either a complemented copy of $\mathbb{R}^ω$ or a complemented copy of the space $(c_{0})_{p}=\{(x_n)_{n\inω}\in \mathbb{R}^ω\colon x_n\to 0\}$, both endowed with the product topology. We show that the latter case holds always when $X\times Y$ is pseudocompact. On the other hand, assuming the Continuum Hypothesis (or even a weaker set-theoretic assumption), we provide an example of a pseudocompact space $X$ such that $C_{p}(X\times X)$ does not contain a complemented copy of $(c_{0})_{p}$. As a corollary to the first result, we show that for all infinite Tychonoff spaces $X$ and $Y$ the space $C_{p}(X\times Y)$ is linearly homeomorphic to the space $C_{p}(X\times Y)\times\mathbb{R}$, although, as proved earlier by Marciszewski, there exists an infinite compact space $X$ such that $C_{p}(X)$ cannot be mapped onto $C_{p}(X)\times\mathbb{R}$ by a continuous linear surjection. This provides a positive answer to a problem of Arkhangel'ski for spaces of the form $C_p(X\times Y)$. Another corollary asserts that for every infinite Tychonoff spaces $X$ and $Y$ the space $C_{k}(X\times Y)$ of continuous functions on $X\times Y$ endowed with the compact-open topology admits a quotient map onto a space isomorphic to one of the following three spaces: $\mathbb{R}^ω$, $(c_{0})_{p}$ or $c_{0}$.

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arXiv2022

Locally compact, $ω_1$-compact spaces

An $ω_1$-compact space is a space in which every closed discrete subspace is countable. We give various general conditions under which a locally compact, $ω_1$-compact space is $σ$-countably compact, i.e., the union of countably many countably compact spaces. These conditions involve very elementary properties. Many results shown here are independent of the usual (ZFC) axioms of set theory, and the consistency of some may involve large cardinals. For example, it is independent of the ZFC axioms whether every locally compact, $ω_1$-compact space of cardinality $\aleph_1$ is $σ$-countably compact. Whether $\aleph_1$ can be replaced with $\aleph_2$ is a difficult unsolved problem. Modulo large cardinals, it is also ZFC-independent whether every hereditarily normal, or every monotonically normal, locally compact, $ω_1$-compact space is $σ$-countably compact. As a result, it is also ZFC-independent whether there is a locally compact, $ω_1$-compact Dowker space of cardinality $\aleph_1$, or one that does not contain both an uncountable closed discrete subspace and a copy of the ordinal space $ω_1$. Set theoretic tools used for the consistency results include the existence of a Souslin tree, the Proper Forcing Axiom (PFA), and models generically referred to as ``MM(S)[S]''. Most of the work is one by the $P$-Ideal Dichotomy (PID) axiom, which holds in the latter two cases, and which requires no large cardinal axioms when directly applied to topological spaces of cardinality $\aleph_1$, as it is in several theorems.

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arXiv2022

On well-splitting posets

We introduce a class of proper posets which is preserved under countable support iterations, includes $ω^ω$-bounding, Cohen, Miller, and Mathias posets associated to filters with the Hurewicz covering properties, and has the property that the ground model reals remain splitting and unbounded in corresponding extensions. Our results may be considered as a possible path towards solving variations of the famous Roitman problem.

math.LO↗
arXiv2021

On regular separable countably compact $\mathbb{R}$-rigid spaces

A topological space $X$ is said to be {\em $Y$-rigid} if any continuous map $f:X\rightarrow Y$ is constant. In this paper we construct a number of examples of regular countably compact $\mathbb R$-rigid spaces with additional properties like separability and first countability. This way we answer several questions of Tzannes, Banakh, Ravsky, as well as get a consistent example of $\mathbb R$-rigid Nyikos space. Also, we show that it is consistent with ZFC that for every cardinal $κ<\mathfrak c$ there exists a regular separable countably compact space $X$ which is $Y$-rigid with respect to any $T_1$ space $Y$ of pseudocharacter $\leqκ$.

math.GN↗
arXiv2021

The Josefson--Nissenzweig theorem, Grothendieck property, and finitely supported measures on compact spaces

The celebrated Josefson-Nissenzweig theorem implies that for a Banach space $C(K)$ of continuous real-valued functions on an infinite compact space $K$ there exists a sequence of Radon measures $\langleμ_n\colon\ n\inω\rangle$ on $K$ which is weakly* convergent to the zero measure on $K$ and such that $\big\|μ_n\big\|=1$ for every $n\inω$. We call such a sequence of measures \textit{a Josefson-Nissenzweig sequence}. In this paper we study the situation when the space $K$ admits a Josefson-Nissenzweig sequence of measures such that its every element has finite support. We prove among the others that $K$ admits such a Josefson-Nissenzweig sequence if and only if $C(K)$ does not have the Grothendieck property restricted to functionals from the space $\ell_1(K)$. We also investigate miscellaneous analytic and topological properties of finitely supported Josefson-Nissenzweig sequences on general Tychonoff spaces. We prove that various properties of compact spaces guarantee the existence of finitely supported Josefson-Nissenzweig sequences. One such property is, e.g., that a compact space can be represented as the limit of an inverse system of compact spaces based on simple extensions. An immediate consequence of this result is that many classical consistent examples of Efimov spaces, i.e. spaces being counterexamples to the famous Efimov problem, admit such sequences of measures. Similarly, we show that if $K$ and $L$ are infinite compact spaces, then their product $K\times L$ always admits a finitely supported Josefson--Nissenzweig sequence. As a corollary we obtain a constructive proof that the space $C_p(K\times L)$ contains a complemented copy of the space $c_0$ endowed with the pointwise topology--this generalizes results of Cembranos and Freniche. Finally, we provide a direct proof of the Josefson-Nissenzweig theorem for the case of Banach spaces $C(K)$.

math.FA↗
arXiv2021

Minimally generated Boolean algebras and the Nikodym property

A Boolean algebra $\mathcal A$ has the Nikodym property if every pointwise bounded sequence of bounded finitely additive measures on $\mathcal A$ is uniformly bounded. Assuming the Diamond Principle $\Diamond$, we will construct an example of a minimally generated Boolean algebra $\mathcal A$ with the Nikodym property. The Stone space of such an algebra must necessarily be an Efimov space. The converse is, however, not true - again under $\Diamond$ we will provide an example of a minimally generated Boolean algebra whose Stone space is Efimov but which does not have the Nikodym property. The results have interesting measure-theoretic and topological consequences.

math.FA↗
arXiv2020

Finite powers and products of Menger sets

We construct, using mild combinatorial hypotheses, a real Menger set that is not Scheepers, and two real sets that are Menger in all finite powers, with a non-Menger product. By a forcing-theoretic argument, we show that the same holds in the Blass--Shelah model for arbitrary values of the ultrafilter and dominating number.

math.GN↗
arXiv2019

Selection Principles in the Laver, Miller, and Sacks models

This article is devoted to the interplay between forcing with fusion and combinatorial covering properties. We discuss known instances of this interplay as well as present a new one, namely that in the Laver model for the consistency of the Borel's conjecture, the Hurewicz property is preserved by finite products of metrizable spaces.

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