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Taras Banakh

Publications and source records attributed to Taras Banakh.

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

Metric characterizations of some subsets of the real line

A metric space $(X,d)$ is called a $subline$ if every 3-element subset $T$ of $X$ can be written as $T=\{x,y,z\}$ for some points $x,y,z$ such that $d(x,z)=d(x,y)+d(y,z)$. By a classical result of Menger, every subline of cardinality $\ne 4$ is isometric to a subspace of the real line. A subline $(X,d)$ is called an $n$-$subline$ for a natural number $n$ if for every $c\in X$ and positive real number $r\in d[X^2]$, the sphere $S(c;r):=\{x\in X:d(x,c)=r\}$ contains at least $n$ points. We prove that every $2$-subline is isometric to some additive subgroup of the real line. Moreover, for every subgroup $G\subseteq\mathbb R$, a metric space $(X,d)$ is isometric to $G$ if and only if $X$ is a $2$-subline with $d[X^2]=G_+:= G\cap[0,\infty)$. A metric space $(X,d)$ is called a $ray$ if $X$ is a $1$-subline and $X$ contains a point $o\in X$ such that for every $r\in d[X^2]$ the sphere $S(o;r)$ is a singleton. We prove that for a subgroup $G\subseteq\mathbb Q$, a metric space $(X,d)$ is isometric to the ray $G_+$ if and only if $X$ is a ray with $d[X^2]=G_+$. A metric space $X$ is isometric to the ray $\mathbb R_+$ if and only if $X$ is a complete ray such that $\mathbb Q_+\subseteq d[X^2]$. On the other hand, the real line contains a dense ray $X\subseteq\mathbb R$ such that $d[X^2]=\mathbb R_+$.

math.MG↗
arXiv2023

Examples of strongly rigid countable (semi)Hausdorff spaces

A topological space $X$ is $strongly$ $rigid$ if each non-constant continuous map $f:X\to X$ is the identity map of $X$. A Hausdorff topological space $X$ is called $Brown$ if for any nonempty open sets $U,V\subseteq X$ the intersection $\bar U\cap\bar V$ is infinite. We prove that every second-countable Brown Hausdorff space $X$ admits a stronger topology $τ'$ such that $X'=(X,τ')$ is a strongly rigid anticompact Brown space.This construction yields an example of a countable anticompact Hausdorff space $X$ which is strongly rigid, which answers two problems posed at MathOverflow. By the same method we construct a strongly rigid $k_2$-metrizable semi-Hausdorff space containing a non-closed compact subset, which answers two other problem posed at MathOverflow.

math.GN↗
arXiv2023

Absolutely closed semigroups

Let $\mathcal C$ be a class of topological semigroups. A semigroup $X$ is called $absolutely$ $\mathcal C$-$closed$ if for any homomorphism $h:X\to Y$ to a topological semigroup $Y\in\mathcal C$, the image $h[X]$ is closed in $Y$. Let $\mathsf{T_{\!1}S}$, $\mathsf{T_{\!2}S}$, and $\mathsf{T_{\!z}S}$ be the classes of $T_1$, Hausdorff, and Tychonoff zero-dimensional topological semigroups, respectively. We prove that a commutative semigroup $X$ is absolutely $\mathsf{T_{\!z}S}$-closed if and only if $X$ is absolutely $\mathsf{T_{\!2}S}$-closed if and only if $X$ is chain-finite, bounded, group-finite and Clifford+finite. On the other hand, a commutative semigroup $X$ is absolutely $\mathsf{T_{\!1}S}$-closed if and only if $X$ is finite. Also, for a given absolutely $\mathcal C$-closed semigroup $X$ we detect absolutely $\mathcal C$-closed subsemigroups in the center of $X$.

math.GN↗
arXiv2023

Subgroups of categorically closed semigroups

Let $\mathcal C$ be a class of topological semigroups. A semigroup $X$ is called (1) $\mathcal C$-$closed$ if $X$ is closed in every topological semigroup $Y\in\mathcal C$ containing $X$ as a discrete subsemigroup, (2) $ideally$ $\mathcal C$-$closed$ if for any ideal $I$ in $X$ the quotient semigroup $X/I$ is $\mathcal C$-closed; (3) $absolutely$ $\mathcal C$-$closed$ if for any homomorphism $h:X\to Y$ to a topological semigroup $Y\in\mathcal C$, the image $h[X]$ is closed in $Y$, (4) $injectively$ $\mathcal C$-$closed$ (resp. $\mathcal C$-$discrete$) if for any injective homomorphism $h:X\to Y$ to a topological semigroup $Y\in\mathcal C$, the image $h[X]$ is closed (resp. discrete) in $Y$. Let $\mathsf{T_{\!z}S}$ be the class of Tychonoff zero-dimensional topological semigroups. For a semigroup $X$ let $V\!E(X)$ be the set of all viable idempotents of $X$, i.e., idempotents $e$ such that the complement $X\setminus\frac{H_e}e$ of the set $\frac{H_e}e=\{x\in X:xe=ex\in H_e\}$ is an ideal in $X$. We prove the following results: (i) for any ideally $\mathsf{T_{\!z}S}$-closed semigroup $X$ each subgroup of the center $Z(X)=\{z\in X:\forall x\in X\;\;(xz=zx)\}$ is bounded; (ii) for any $\mathsf{T_{\!z}S}$-closed semigroup $X$, each subgroup of the ideal center $I\!Z(X)=\{z\in Z(X):zX\subseteq Z(X)\}$ is bounded; (iii) for any $\mathsf{T_{\!z}S}$-discrete or injectively $\mathsf{T_{\!z}S}$-closed semigroup $X$, every subgroup of $Z(X)$ is finite, (iv) for any viable idempotent $e$ in an ideally (and absolutely) $\mathsf{T_{\!z}S}$-closed semigroup $X$, the maximal subgroup $H_e$ is ideally (and absolutely) $\mathsf{T_{\!z}S}$-closed and has bounded (and finite) center $Z(H_e)$.

math.GR↗
arXiv2022

Categorically closed countable semigroups

In this paper we establish a connection between categorical closedness and topologizability of semigroups. In particular, for a class $\mathsf T_{\!1}\mathsf S$ of $T_1$ topological semigroups we prove that a countable semigroup $X$ with finite-to-one shifts is injectively $\mathsf T_{\!1}\mathsf S$-closed if and only if $X$ is $\mathsf{T_{\!1}S}$-nontopologizable in the sense that every $T_1$ semigroup topology on $X$ is discrete. Moreover, a countable cancellative semigroup $X$ is absolutely $\mathsf T_{\!1}\mathsf S$-closed if and only if every homomorphic image of $X$ is $\mathsf T_{\!1}\mathsf S$-nontopologizable. Also, we introduce and investigate a notion of a polybounded semigroup. It is proved that a countable semigroup $X$ with finite-to-one shifts is polybounded if and only if $X$ is $\mathsf T_{\!1}\mathsf S$-closed if and only if $X$ is $\mathsf T_{\!z}\mathsf S$-closed, where $\mathsf T_{\!z}\mathsf S$ is a class of zero-dimensional Tychonoff topological semigroups. We show that polyboundedness provides an automatic continuity of the inversion in $T_1$ paratopological groups and prove that every cancellative polybounded semigroup is a group.

math.GN↗
arXiv2022

A non-polybounded absolutely closed $36$-Shelah group

For every infinite cardinal $κ$ with $κ^+=2^κ$ we construct a group $G$ of cardinality $|G|=κ^+$ such that (i) $G$ is $36$-Shelah, which means that $A^{36}=G$ for any subset $A\subseteq G$ of cardinality $|A|=|G|$; (ii) $G$ is absolutely $\mathsf{T_{\!1}S}$-closed and projectively $\mathsf{T_{\!1}S}$-discrete, which means that for every homomorphism $h:G\to Y$ to a $T_1$ topological semigroup $Y$ the image $h[G]$ is a closed discrete subspace of $Y$, (iii) $G$ cannot be covered by finitely many algebraic subsets, i.e., subsets of the form $\{x\in G:xc_1xc_2\cdots xc_n=e\}$ for some $c_1,c_2,\cdots,c_n\in G$.

math.GR↗
arXiv2022

Polyboundedness of zero-closed semigroups

The polyboundedness number $\mathrm{cov}(\mathcal A_X)$ of a semigroup $X$ is the smallest cardinality of a cover of $X$ by sets of the form $\{x\in X:a_0xa_1\cdots xa_n=b\}$ for some $n\ge 1$, $b\in X$ and $a_0,\dots,a_n\in X^1=X\cup\{1\}$. Semigroups with finite polyboundedness number are called polybounded. A semigroup $X$ is called zero-closed if $X$ is closed in its $0$-extension $X^0=\{0\}\cup X$ endowed with any Hausdorff semigroup topology. We prove that any zero-closed infinite semigroup $X$ has $\mathrm{cov}(\mathcal A_X)<|X|$. Under Martin's Axiom, a zero-closed semigroup is polybounded if $X$ admits a compact Hausdorff semigroup topology or $X$ has a separable complete subinvariant metric.

math.GR↗
arXiv2022

First-countable Lindelöf scattered spaces

We study the class of first-countable Lindelöf scattered spaces, or "FLS" spaces. While every $T_3$ FLS space is homeomorphic to a scattered subspace of $\mathbb Q$, the class of $T_2$ FLS spaces turns out to be surprisingly rich. Our investigation of these spaces reveals close ties to $Q$-sets, Lusin sets, and their relatives, and to the cardinals $\mathfrak{b}$ and $\mathfrak{d}$. Many natural questions about FLS spaces turn out to be independent of $\mathsf{ZFC}$. We prove that there exist uncountable FLS spaces with scattered height $ω$. On the other hand, an uncountable FLS space with finite scattered height exists if and only if $\mathfrak{b} = \aleph_1$. We prove some independence results concerning the possible cardinalities of FLS spaces, and concerning what ordinals can be the scattered height of an FLS space. Several open problems are included.

math.GN↗
arXiv2022

$E$-separated semigroups

A semigroup is called $E$-$separated$ if for any distinct idempotents $x,y\in X$ there exists a homomorphism $h:X\to Y$ to a semilattice $Y$ such that $h(x)\ne h(y)$. Developing results of Putcha and Weissglass, we characterize $E$-separated semigroups via certain commutativity properties of idempotents of $X$. Also we characterize $E$-separated semigroups in the class of $π$-regular $E$-semigroups.

math.RA↗
arXiv2022

Categorically closed unipotent semigroups

Let $\mathcal C$ be a class of $T_1$ topological semigroups, containing all Hausdorff zero-dimensional topological semigroups. A semigroup $X$ is $\mathcal C$-$closed$ if $X$ is closed in any topological semigroup $Y\in\mathcal C$ that contains $X$ as a discrete subsemigroup; $X$ is $injectively$ $\mathcal C$-$closed$ if for any (injective) homomorphism $h:X\to Y$ to a topological semigroup $Y\in\mathcal C$, the image $h[X]$ is closed in $Y$. A semigroup $X$ is $unipotent$ if it contains a unique idempotent. We prove that a unipotent commutative semigroup $X$ is (injectively) $\mathcal C$-closed if and only if $X$ is bounded, nonsingular (and group-finite). This characterization implies that for every injectively $\mathcal C$-closed unipotent semigroup $X$, the center $Z(X)$ is injectively $\mathcal C$-closed.

math.GN↗
arXiv2022

Injectively and absolutely $T_1S$-closed semigroups

A semigroup $X$ is $absolutely$ (resp. $injectively$) $T_1S$-$closed$ if for any (injective) homomorphism $h:X\to Y$ to a $T_1$ topological semigroup $Y\in\mathcal C$, the image $h[X]$ is closed in $Y$. We prove that a commutative semigroup $X$ is injectively $T_1S$-closed if and only if $X$ is bounded, nonsingular and Clifford-finite. Using this characterization, we prove that (1) every injectively $T_1S$-closed semigroup has injectively $T_1S$-closed center, and (2) every absolutely $T_1S$-closed semigroup has finite center.

math.GN↗
arXiv2022

Injectively closed commutative semigroups

Let $\mathcal C$ be a class of topological semigroups. A semigroup $X$ is $injectively$ $\mathcal C$-$closed$ if $X$ is closed in each topological semigroup $Y\in\mathcal C$ containing $X$ as a subsemigroup. Let $\mathsf{T_{\!2}S}$ (resp. $\mathsf{T_{\!z}S}$) be the class of Hausdorff (and zero-dimensional) topological semigroups. We prove that a commutative semigroup $X$ is injectively $\mathsf{T_{\!2}S}$-closed if and only if $X$ is injectively $\mathsf{T_{\!z}S}$-closed if and only if $X$ is bounded, chain-finite, group-finite, nonsingular and not Clifford-singular.

math.GR↗
arXiv2022

Automatic continuity of measurable homomorphisms on Cech-complete topological groups

We prove that a homomorphism $h:X\to Y$ from a (locally compact) Cech-complete topological group $X$ to a topological group $Y$ is continuous if and only if $h$ is Borel-measurable if and only if $h$ is universally measurable (if and only if $h$ is Haar-measurable). This answers a problem of Kuznetsova and extends a result of Kleppner on the continuity of Haar-measurable homomorphisms between locally compact groups and a result of Rosendal on the continuity of universally measurable homomorphisms between Polish groups.

math.GN↗
arXiv2022

The Set-Cover game and nonmeasurable unions

Using a game-theoretic approach we present a generalization of the classical result of Brzuchowski, Cichoń, Grzegorek and Ryll-Nardzewski on non-measurable unions. We also present applications of obtained results to Marczewski--Burstin representable ideals, as well as to establishing some countability and continuity properties of measurable functions and homomorphisms between topological groups.

math.GN↗
arXiv2022

$Q$-spaces, perfect spaces and related cardinal characteristics of the continuum

A topological space $X$ is called a $Q$-space if every subset of $X$ is of type $F_σ$ in $X$. For $i\in\{1,2,3\}$ let $\mathfrak q_i$ be the smallest cardinality of a second-countable $T_i$-space which is not a $Q$-space. It is clear that $\mathfrak q_1\le\mathfrak q_2\le\mathfrak q_3$. For $i\in\{1,2\}$ we prove that $\mathfrak q_i$ is equal to the smallest cardinality of a second-countable $T_i$-space which is not perfect. Also we prove that $\mathfrak q_3$ is equal to the smallest cardinality of a submetrizable space, which is not a $Q$-space. Martin's Axiom implies that $\mathfrak q_i=\mathfrak c$ for all $i\in\{1,2,3\}$.

math.GN↗
arXiv2022

On symmetrizability and perfectness of second-countable spaces

A symmetrizability criterion of Arhangelskii implies that a second-countable Hausdorff space is symmetrizable if and only if it is perfect. We present an example of a non-symmetrizable second-countable submetrizable space of cardinality $\mathfrak q_0$ and study the smallest possible cardinality $\mathfrak q_i$ of a non-symmetrizable second-countable $T_i$-space for $i\in\{1,2\}$.

math.GN↗
arXiv2022

The binary quasiorder on semigroups

Given two elements $x,y$ of a semigroup $X$ we write $x\lesssim y$ if for every homomorphism $χ:X\to\{0,1\}$ we have $χ(x)\leχ(y)$. The quasiorder $\lesssim$ is called the $binary$ $quasiorder$ on $X$. It induces the equivalence relation $\Updownarrow$ that coincides with the least semilattice congruence on $X$. In the paper we discuss some known and new properties of the binary quasiorder on semigroups.

math.GR↗