# Taras BanakhIvan Franko National University of Lviv | ivan franko · Department of Mathematics

Taras Banakh

Dr.Sci., Prof.

## About

510

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Introduction

Education

September 1984 - June 1989

## Publications

Publications (510)

Let C be a class of T1 topological semigroups, containing all Hausdorff zero-dimensional topological semigroups. A semigroup X is C-closed if X is closed in any topological semigroup Y∈C that contains X as a discrete subsemigroup; X is injectively C-closed if for any injective homomorphism h:X→Y to a topological semigroup Y∈C the image h[X] is clos...

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...

In this paper we are interested in finding and evaluating cardinal characteristics of the continuum that appear in large-scale topology, usually as the smallest weights of coarse structures that belong to certain classes (indiscrete, inseparable, large) of finitary or locally finite coarse structures on ω. Besides well-known cardinals b,d,c we shal...

We study the class of first-countable Lindelöf scattered spaces, or “FLS” spaces. While every T3 FLS space is homeomorphic to a scattered subspace of Q, the class of T2 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 b and d. Many nat...

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) $absolutely$ $...

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 c...

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 c...

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) homomor...

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) $projectively$ $\mathcal C$-$closed$ if every homomorphic image of $X$ is $\mathcal C$-closed, (3) $ideally$ $\mathcal C$-$closed$...

We study the class of first-countable Lindel\"of 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 cardina...

We define a topological ring $R$ to be Hirsch, if for any unconditionally convergent series $\sum_{n\in\omega} x_i$ in $R$ and any neighborhood $U$ of the additive identity $0$ of $R$ there exists a neighborhood $V\subseteq R$ of $0$ such that $\sum_{n\in F} a_n x_n\in U$ for any finite set $F\subset\omega$ and any sequence $(a_n)_{n\in F}\in V^F$....

We prove that a homomorphism $h:X\to Y$ from a (locally compact) \v 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 con...

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-countab...

A topological space $X$ is called a $Q$-space if every subset of $X$ is of type $F_\sigma$ 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...

We study Vietoris hyperspaces of closed and final sets of Priestley spaces. We are particularly interested in Skula topologies. A topological space is Skula if its topology is generated by differences of open sets of another topology. A compact Skula space is scattered and moreover has a natural well-founded ordering compatible with the topology, n...

Several new characterizations of the Gelfand–Phillips property are given. We define a strong version of the Gelfand–Phillips property and prove that a Banach space has this stronger property iff it embeds into c0. For an infinite compact space K, the Banach space C(K) has the strong Gelfand–Phillips property iff C(K) is isomorphic to c0 iff K is co...

The extent $e(X)$ of a topological space $X$ is the supremum of sizes of closed discrete subspaces of $X$. Assuming that $X$ belongs to some class of topological spaces, we bound $e(X)$ byother cardinal characteristics of $X$, for instance Lindel\"of number, spread or density.

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 se...

We prove that for any K-analytic subsets A,B of a locally compact group X if the product AB has empty interior (and is meager) in X, then one of the sets A or B can be covered by countably many closed nowhere dense subsets (of Haar measure zero) in X. This implies that a K-analytic subset A of X can be covered by countably many closed Haar-null set...

Given two elements $x,y$ of a semigroup $X$ we write $x\lesssim y$ if for every homomorphism $\chi:X\to\{0,1\}$ we have $\chi(x)\le\chi(y)$. 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 som...

We prove that the countable product of lines contains a Borel linear subspace $L\ne\mathbb R^\omega$ that cannot be covered by countably many closed Haar-meager sets. This example is applied to studying the interplay between various classes of ``large'' sets and Kuczma--Ger classes in the topological vector spaces $\mathbb R^n$ for $n\le \omega$.

Let A,X,Y be Banach spaces and A×X→Y, (a,x)↦ax be a continuous bilinear function, called a Banach action. We say that this action preserves unconditional convergence if for every bounded sequence (an)n∈ω in A and unconditionally convergent series ∑n∈ωxn in X, the series ∑n∈ωanxn is unconditionally convergent in Y. We prove that a Banach action A×X→...

Answering a question of Garbuli\'nska-W\c{e}grzyn and Kubi\'s, we prove that Gurarii operators form a dense $G_\delta$-set in the space $\mathcal B(\mathbb G)$ of all nonexpansive operators on the Gurarii space $\mathbb G$, endowed with the strong operator topology. This implies that the set of universal operators on $\mathbb G$ form a residual set...

We prove that for any coarse spaces X1,…,Xn of asymptotic dimension ≥1, the product X=X1×…×Xn has asymptotic dimension ≥n. Another result states that a finitary coarse space Z has asdim(Z)≥n if Z admits an almost free action of the group Zn. We deduce these inequalities from the following combinatorial result (that generalizes the Hex Theorem of Ga...

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 $\mathsf{T_{\!1}S}$-...

Let $A,X,Y$ be Banach spaces and $A\times X\to Y$, $(a,x)\mapsto ax$, be a continuous bilinear function, called a *Banach action*. We say that this action *preserves unconditional convergence* if for every bounded sequence $(a_n)_{n\in\omega}$ in $A$ and unconditionally convergent series $\sum_{n\in\omega}x_n$ in $X$ the series $\sum_{n\in\omega}a_...

We extend the well-known Gelfand-Phillips property for Banach spaces to locally convex spaces, defining a locally convex space $E$ to be Gelfand-Phillips if every limited set in $E$ is precompact in the topology on $E$ defined by barrels. Several characterizations of Gelfand-Phillips spaces are given. The problem of preservation of the Gelfand-Phil...

We introduce the strong Gelfand-Phillips property for locally convex spaces and give several characterizations of this property. We characterize the strong Gelfand-Phillips property among locally convex spaces admitting a stronger Banach space topology. If $C_{\mathcal T}(X)$ is a space of continuous functions on a Tychonoff space $X$, endowed with...

A Hausdorff topological space X is called superconnected (resp. coregular) if for any nonempty open sets U1,…Un⊆X, the intersection of their closures U‾1∩…∩U‾n is not empty (resp. the complement X∖(U‾1∩…∩U‾n) is a regular topological space). A canonical example of a coregular superconnected space is the projective space QP∞ of the topological vecto...

Let $G$ be a paratopological group.Following F.~Lin and S.~Lin, we say that the group $G$ is pseudobounded,if for any neighborhood $U$ of the identity of $G$,there exists a natural number $n$ such that $U^n=G$.The group $G$ is $\omega$-pseudobounded,if for any neighborhood $U$ of the identity of $G$, the group $G$ is aunion of sets $U^n$, where $n$...

Under Jensen's diamond principle $\diamondsuit$, we construct a simple Efimov space $K$ whose space of nonatomic probability measures $P_{na}(K)$ is first-countable and sequentially compact. These two properties of $P_{na}(K)$ imply that the space of probability measures $P(K)$ on $K$ is selectively sequentially pseudocompact and the Banach space $...

Several new characterizations of the Gelfand-Phillips property are given. We define a strong version of the Gelfand-Phillips property and prove that a Banach space has this stronger property iff it embeds into $c_0$. For an infinite compact space $K$, the Banach space $C(K)$ has the strong Gelfand-Phillips property iff $C(K)$ is isomorphic to $c_0$...

Let $G$ be a paratopological group. Following F. Lin and S. Lin, we say that the group $G$ is pseudobounded, if for any neighborhood $U$ of the identity of $G$, there exists a natural number $n$ such that $U^n=G$. The group $G$ is $\omega$-pseudobounded, if for any neighborhood $U$ of the identity of $G$, the group $G$ is a union of sets $U^n$, whe...

We prove that every isometry between the unit spheres of 2-dimensional Banach spaces extends to a linear isometry of the Banach spaces. This resolves the famous Tingley's problem in the class of 2-dimensional Banach spaces.

Let C be a class of T1 topological semigroups which contains all zero-dimensional Hausdorff topological semigroups. A semigroup X is called C-closed if X is closed in each topological semigroup Y∈C containing X as a discrete subsemigroup; X is projectively C-closed if for each congruence ≈ on X the quotient semigroup X/≈ is C-closed. A semigroup X...

For a Hausdorff topologized semilattice X its Lawson numberΛ¯(X) is the smallest cardinal κ such that for any distinct points x,y∈X there exists a family U of closed neighborhoods of x in X such that |U|≤κ and ⋂U is a subsemilattice of X that does not contain y. It follows that Λ¯(X)≤ψ¯(X), where ψ¯(X) is the smallest cardinal κ such that for any p...

The Golomb space (resp. the Kirch space) is the set N of positive integers endowed with the topology generated by the base consisting of arithmetic progressions a+bN0={a+bn:n≥0} where a,b∈N and b is a (square-free) number, coprime with a. It is known that the Golomb space (resp. the Kirch space) is connected (and locally connected). By a recent res...

Given a T0 paratopological group G and a class C of continuous homomorphisms of paratopological groups, we define the C-semicompletion C[G) and C-completion C[G] of the group G that contain G as a dense subgroup, satisfy the T0-separation axiom and have certain universality properties. For special classes C, we present some necessary and sufficient...

For any metric space $X$ endowed with the action of a group $G$, and two $n$-gons $\vec x=(x_1,\dots,x_n)\in X^n$ and $\vec y=(y_1,\dots,y_n)\in X^n$ in $X$, we introduce the $G$-deviation $d(G\vec x,\vec y\,)$ of $\vec x$ from $\vec y$ as the distance in $X^n$ from $\vec y$ to the $G$-orbit $G\vec x$ of $\vec x$ in the $n$-th power $X^n$ of $X$. F...

A subset $B$ of a group $G$ is called a basis of $G$ if each element $g\in G$ can be written as $g=ab$ for some elements $a,b\in B$. The smallest cardinality $|B|$ of a basis $B\subseteq G$ is called the basis size of $G$ and is denoted by $r[G]$. We prove that each finite group $G$ has $r[G]>\sqrt{|G|}$. If $G$ is Abelian, then $r[G]\ge \sqrt{2|G|...

For any metric space $X$ endowed with the action of a group $G$, and two $n$-gons $\vec x=(x_1,\dots,x_n)\in X^n$ and $\vec y=(y_1,\dots,y_n)\in X^n$ in $X$, we introduce the $G$-deviation $d(G\vec x,\vec y\,)$ of $\vec x$ from $\vec y$ as the distance in $X^n$ from $\vec y$ to the $G$-orbit $G\vec x$ of $\vec x$ in the $n$-th power $X^n$ of $X$. F...

A locally convex space (lcs) E is said to have an ωω-base if E has a neighborhood base {Uα:α∈ωω} at zero such that Uβ⊆Uα for all α≤β. The class of lcs with an ωω-base is large, among others contains all (LM)-spaces (hence (LF)-spaces), strong duals of distinguished Fréchet lcs (hence spaces of distributions D′(Ω)). A remarkable result of Cascales-O...

Given a topological ring $R$, we study semitopological $R$-modules, construct their completions, Bohr and borno modifications. For every topological space $X$, we construct the free (semi)topological $R$-module over $X$ and prove that for a $k$-space $X$ its free semitopological $R$-module is a topological $R$-module. Also we construct a Tychonoff...

A Banach space X has the Mazur–Ulam property if any isometry from the unit sphere of X onto the unit sphere of any other Banach space Y extends to a linear isometry of the Banach spaces X,Y. A Banach space X is called smooth if the unit ball has a unique supporting functional at each point of the unit sphere. We prove that each non-smooth 2-dimensi...

We prove that every isometry between the unit spheres of 2-dimensional Banach spaces extends to a linear isometry of the Banach spaces. This resolves the famous Tingley's problem in the class of 2-dimensional Banach spaces.

A Banach space $X$ has the Mazur-Ulam property if any isometry from the unit sphere of $X$ onto the unit sphere of any other Banach space $Y$ extends to a linear isometry of the Banach spaces $X,Y$.
A Banach space $X$ is called smooth if the unit ball has a unique supporting functional at each point of the unit sphere. We prove that each non-smooth...

A 2-dimensional Banach space X is called absolutely smooth if its unit sphere is the image of the real line under a differentiable function r:R→SX whose derivative is locally absolutely continuous and has ‖r′(s)‖=1 for all s∈R. We prove that any isometry f:SX→SY between the unit spheres of absolutely smooth Banach spaces X,Y extends to a linear iso...

Let $M$ be an ANR space and $X$ be a homotopy dense subspace in $M$. Assume that $M$ admits a continuous binary operation $* : M \times M \to M$ such that for every $x, y \in M$ the inclusion $x * y \in X$ holds if and only if $x, y \in X$. Assume also that there exist continuous unary operations $u, v : M \to M$ such that $x = u(x) * v(x)$ for all...

We prove that every usco multimap Φ:X→Y from a metrizable separable space X to a GO-space Y has an Fσ-measurable selection. On the other hand, for the split interval I¨ and the projection P:I¨2→I2 of its square onto the unit square I2, the usco multimap P-1:I2⊸I¨2 has a Borel (Fσ-measurable) selection if and only if the Continuum Hypothesis holds....

For a topological space X its reflection in a class T of topological spaces is a pair (TX,iX) consisting of a space TX∈T and a continuous map iX:X→TX such that for any continuous map f:X→Y to a space Y∈T there exists a unique continuous map f¯:TX→Y such that f=f¯∘iX. In this paper for an infinite cardinal κ and a nonempty set M of ultrafilters on κ...

Let C be a class of Hausdorff topological semigroups which contains all zero-dimensional Hausdorff topological semigroups. A semigroup X is called C-closed if X is closed in each topological semigroup Y ∈ C containing X as a discrete subsemigroup; X is projectively C-closed if for each congruence ≈ on X the quotient semigroup X/≈ is C-closed. A sem...

A subset A of a semigroup S is called a chain (antichain) if ab∈{a,b} (ab∉{a,b}) for any (distinct) elements a,b∈A. A semigroup S is called periodic if for every element x∈S there exists n∈N such that xn is an idempotent. A semigroup S is called (anti)chain-finite if S contains no infinite (anti)chains. We prove that each antichain-finite semigroup...

We discuss various modifications of separability, precompactness and narrowness in topological groups and test those modifications in the permutation groups S(X) and \(S_{<\omega }(X)\).

Given a topological ring R, we study semitopological R-modules, construct their completions, Bohr and borno modications. For every topological space X, we construct the free (semi)topological R-module over X and prove that for a k-space X its free semitopological R-module is a topological R-module. Also we construct a Tychono space X whose free sem...

Given a T_0 paratopological group G and a class C of continuous homomor-phisms of paratopological groups, we define the C-semicompletion C[G) and C-completion C[G] of the group G that contain G as a dense subgroup, satisfy the T_0-separation axiom and have certain universality properties. For special classes C, we present some necessary and suffici...

A subset $A$ of a semigroup $S$ is called a chain (antichain) if $xy \in \{x, y\}$ ($xy \notin \{x, y\}$) for any (distinct) elements $x, y \in S$. A semigroup $S$ is called (anti)chain-finite if $S$ contains no infinite (anti)chains. We prove that each antichain-finite semigroup S is periodic and for every idempotent e of S the set $\{x \in S : \e...

A topological group X is called duoseparable if there exists a countable set S⊆X such that SUS=X for any neighborhood U⊆X of the identity. We construct a functor F assigning to each (abelian) topological group X a duoseparable (abelian-by-cyclic) topological group FX, containing an isomorphic copy of X. In fact, the functor F is defined on the cate...

A function f : X → Y between topological spaces is called σ-continuous (resp. ̄σ-continuous ) if there exists a (closed) cover { X n } n ∈ ω of X such that for every n ∈ ω the restriction f ↾ X n is continuous. By 𝔠 σ (resp. 𝔠 ¯σ )we denote the largest cardinal κ ≤ 𝔠 such that every function f : X → ℝ defined on a subset X ⊂ ℝ of cardinality | X |...

Using a game-theoretic approach we present a generalization of the classical result of Brzuchowski, Cicho\'n, 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 funct...

We construct a metrizable Lawson semitopological semilattice $X$ whose partial order $\le_X\,=\{(x,y)\in X\times X:xy=x\}$ is not closed in $X\times X$. This resolves a problem posed earlier by the authors.

We obtain many results and solve some problems about feebly compact paratopological groups. We obtain necessary and sufficient conditions for such a group to be topological. One of them is the quasiregularity. We prove that each 2-pseudocompact paratopological group is feebly compact and that each Hausdorff σ-compact feebly compact paratopological...

For any compact Hausdorff space K we construct a canonical finitary coarse structure EX,K on the set X of isolated points of K. This construction has two properties:
(1)If a finitary coarse space (X,E) is metrizable, then its coarse structure E coincides with the coarse structure EX,K generated by the Higson compactification K of X.
(2)A compact Ha...

We study Vietoris hyperspaces of closed and closed final sets of Priestley spaces. We are particularly interested in Skula topologies. A topological space is \emph{Skula} if its topology is generated by differences of open sets of another topology. A compact Skula space is scattered and moreover has a natural well-founded ordering compatible with t...

A locally convex space (lcs) $E$ is said to have an $\omega^{\omega}$-base if $E$ has a neighborhood base $\{U_{\alpha}:\alpha\in\omega^\omega\}$ at zero such that $U_{\beta}\subseteq U_{\alpha}$ for all $\alpha\leq\beta$. The class of lcs with an $\omega^{\omega}$-base is large, among others contains all $(LM)$-spaces (hence $(LF)$-spaces), strong...

A topological space is Suslin (Lusin) if it is a continuous (and bijective) image of a Polish space. For a Tychonoff space X let \(C_p(X)\), \(C_k(X)\) and \(C_{{\downarrow }{\mathsf {F}}}(X)\) be the space of continuous real-valued functions on X, endowed with the topology of pointwise convergence, the compact-open topology, and the Fell hypograph...

The Golomb space (resp. the Kirch space) is the set $N$ of positive integers endowed with the topology generated by the base consisting of arithmetic progressions $a+bN_0=\{a+bn:n\ge 0\}$ where $a\in N$ and $b$ is a (square-free) number, coprime with $a$. It is known that the Golomb space (resp. the Kirch space) is connected (and locally connected)...

A topologized semilattice X is called complete if each non-empty chain C⊂X\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C\subset X$$\end{document} has infC∈C¯\documen...

A bornology ℬ on a set X is called minmax, if the smallest and largest coarse structures on X compatible with ℬ coincide. We prove that ℬ is minmax, if and only if the family ℬ# = {p ∈ βX : {X\B : B ∈ ℬ} ⊂ p} consists of ultrafilters which are pairwise non-isomorphic via ℬ-preserving bijections of X. In addition, we construct a minmax bornology ℬ o...

Let κ be an infinite cardinal. A topological space X is κ-bounded if the closure of any subset of cardinality ≤κ in X is compact. We discuss the problem of embeddability of topological spaces into Hausdorff (Urysohn, regular) κ-bounded spaces, and present a canonical construction of such an embedding. Also we construct a (consistent) example of a s...

A function f : X → ℝ defined on a topological space X is called returning if for any point x ∈ X there exists a positive real number M x such that for every path-connected subset C x ⊂ X containing the point x and any y ∈ C x ∖ { x } there exists a point z ∈ C x ∖ { x , y } such that | f ( z )| ≤ max{ M x , | f ( y )|}. A topological space X is cal...

Lecture notes of the Introductory course in the Classical Set Theory, based on axioms of von Neumann-Bernays-Godel.

A topological semigroup is monothetic provided it contains a dense cyclic subsemigroup. The Koch problem asks whether every locally compact monothetic monoid is compact. This problem was opened for more than sixty years, till in 2018 Zelenyuk obtained a negative answer. In this paper we obtain a positive answer for Koch’s problem for some special c...

Let $p\in[1,\infty]$ and $F:Set\to Set$ be a functor with finite supports in the category $Set$ of sets. Given a non-empty metric space $(X,d_X)$, we introduce the distance $d^p_{FX}$ on the functor-space $FX$ as the largest distance such that for every $n\in IN$ and $a\in Fn$ the map $X^n\to FX$, $f\mapsto Ff(a)$, is non-expanding with respect to...

For a metric space $X$, let $\mathsf FX$ be the space of all nonempty finite subsets of $X$ endowed with the largest metric $d^1_{\mathsf FX}$ such that for every $n\in\mathbb N$ the map $X^n\to\mathsf FX$, $(x_1,\dots,x_n)\mapsto \{x_1,\dots,x_n\}$, is non-expanding with respect to the $\ell^1$-metric on $X^n$. We study the completion of the metri...

In this paper we collect some open set-theoretic problems that appear in the large-scale topology (called also Asymptology). In particular we ask problems about critical cardinalities of some special (large, indiscrete, inseparated) coarse structures on ω, about the interplay between properties of a coarse space and its Higson corona, about some sp...

Let B be a class of finite-dimensional Banach spaces. A B-decomposed Banach space is a Banach space X endowed with a family BX⊂B of subspaces of X such that each x∈X can be uniquely written as the sum of an unconditionally convergent series ∑B∈BXxB for some (xB)B∈BX∈∏B∈BXB. For every B∈BX let prB:X→B denote the coordinate projection. Let C⊂[-1,1] b...

Assume that a functionally Hausdorff space X is a continuous image of a Čech complete space P with Lindelöf number l(P)<c. Then the following conditions are equivalent: (i) every compact subset of X is scattered, (ii) for every continuous map f:X→Y to a functionally Hausdorff space Y the image f(X) has cardinality not exceeding max{l(P),ψ(Y)}, (ii...

We construct a metrizable semitopological semilattice X whose partial order P = {(x, y) ∈ X × X : xy = x} is a non-closed dense subset of X × X. As a by-product we find necessary and sufficient conditions for the existence of a (metrizable) Hausdorff topology on a set, act, semigroup or semilattice, having a prescribed countable family of convergen...

We prove that an injective map f:X→Y between metrizable spaces X,Y is continuous if for every connected subset C⊂X the image f(C) is connected and one of the following conditions is satisfied:
•Y is a 1-manifold and X is compact and connected;
•Y is a 2-manifold and X is a closed 2-manifold;
•Y is a 3-manifold and X is a rational homology 3-sphere....

We define a locally convex space E to have the Josefson-Nissenzweig property (JNP) if the identity map (E , σ(E , E)) → (E , β * (E , E)) is not sequentially continuous. By the classical Josefson-Nissenzweig theorem, every infinite-dimensional Banach space has the JNP. We show that for a Tychonoff space X, the function space Cp(X) has the JNP iff t...

A Hausdorff topological space $X$ is called {\em superconnected} (resp. {\em coregular}) if for any nonempty open sets $U_1,\dots U_n\subseteq X$, the intersection of their closures $\overline U_1\cap\dots\cap\overline U_n$ is not empty (resp. the complement $X\setminus (\overline U_1\cap\dots\cap\overline U_n)$ is a regular topological space). A c...

Two non-empty sets A, B of a metric space (X, d) are called parallel if \(d(a,B)=d(A,B)=d(A,b)\) for any points \(a\in A\) and \(b\in B\). Answering a question posed on mathoverflow.net, we prove that for a cover \({\mathscr {C}}\) of a metrizable space X by compact subsets, the following conditions are equivalent: (i) the topology of X is generate...

We study the Baire type properties in the classes Bα(X,Y) of Baire-α functions and Bαst(X,Y) of stable Baire-α functions from a topological space X to a topological space Y, where α≥1 is a countable ordinal. Among others we prove the following results. If X is a normal space, then B1(X)=RX iff X is a Q-space. If X is a Tychonoff space of countable...

In this paper we are interested in finding and evaluating cardinal characteristics of the continuum that appear in large-scale topology, usually as the smallest weights of coarse structures that belong to certain classes (indiscrete, inseparated, large) of finitary or locally finite coarse structures on $\omega$. Besides well-known cardinals $\math...

The material of this preprint was included as Section 6 to the paper "Small uncountable cardinals in large-scale topology".

We discuss various modifications of separability, precompactnmess and narrowness in topological groups and test those modifications in the permutation groups $S(X)$ and $S_{<\omega}(X)$.

A topological group $X$ is called $duoseparable$ if there exists a countable set $S\subseteq X$ such that $SUS=X$ for any neighborhood $U\subseteq X$ of the unit. We construct a functor $F$ assigning to each (abelian) topological group $X$ a duoseparable (abelain-by-cyclic) topological group $FX$, containing an isomorphic copy of $X$. In fact, the...

For any compact Hausdorff space $K$ we construct a canonical finitary coarse structure $\mathcal E_{X,K}$ on the set $X$ of isolated points of $K$. This construction has two properties: $\bullet$ If a finitary coarse space $(X,\mathcal E)$ is metrizable, then its coarse structure $\mathcal E$ coincides with the coarse structure $\mathcal E_{X,\bar...

We prove that for any coarse spaces $X_1,\dots,X_n$ of asymptotic dimension $\ge 1$, the product $X=X_1\times\dots\times X_n$ has asymptotic dimension $\ge n$. Another result states that a finitary coare space $Z$ has $asdim (Z)\ge n$ if $Z$ admits an almost free action of the group $\mathbb Z^n$. We deduce these results from the following combinat...

The path component space of a topological space $X$ is the quotient space $\pi_0(X)$ whose points are the path components of $X$. We show that every Tychonoff space $X$ is the path-component space of a Tychonoff space $Y$ of weight $w(Y)=w(X)$ such that the natural quotient map $Y\to \pi_0(Y)=X$ is a perfect map. Hence, many topological properties...

A subset D of an abelian group is decomposable if ∅≠D⊂D+D. In the paper we give partial answers to an open problem asking whether every finite decomposable subset D of an abelian group contains a non-empty subset Z⊂D with ∑Z=0. For every n∈N we present a decomposable subset D of cardinality |D|=n in the cyclic group of order 2n−1 such that ∑D=0, bu...

A bornology $\mathcal{B}$ on a set $X$ is called minmax, if the smallest and largest coarse structures on $X$ compatible with $\mathcal{B}$ coincide. We prove that $\mathcal{B}$ is minmax, if and only if the family $\mathcal B^\sharp=\{p\in\beta X:\{X\setminus B:B\in\mathcal B\}\subset p\}$ consists of ultrafilters which are pairwise non-isomorphic...

The linear continuity of a function defined on a vector space means that its restriction to every affine line is continuous. For functions defined on Rm this notion is close to the separate continuity which requires only continuity on straight lines which are parallel to coordinate axes. The classical Lebesgue theorem states that every separately c...

## Questions

Question (1)

A topological space $X$ is defined to have *countable discrete cellularity* if each discrete family of open subsets of $X$ is at most countable.

A family $\mathcal F$ of subsets of a topological space $X$ is called *discrete* if each point $x\in X$ has a neighborhood $O_x\subset X$ that intersects at most one set $F\in\mathcal F$.

It is easy to see that a Tychonoff space $X$ has countable discrete cellularity if and only if for any continuous map $f:X\to M$ to a metric space $M$ the image $f(X)$ is separable.

A topological group $G$ is *$\omega$-narrow* if for any neighborhood $U$ of the unit there exists a countable subset $C\subset G$ such that $G=C\cdot U$. By a classical theorem of Guran, a topological group is $\omega$-narrow if and only if $G$ is topologically isomorphic to a subgroup of a Tychonoff product of metrizable separable topological groups.

It is easy to see that a topological group is $\omega$-narrow if it has countable discrete cellularity. What about the converse?

>**Problem 1.** Does every $\omega$-narrow topological group have countable discrete cellularity?

This problem can also be asked for uniform spaces. A uniform space $(X,\mathcal U)$ is *$\omega$-narrow* if for any entouage $U\in\mathcal U$ there exists a countable set $C\subset X$ such that for every $x\in X$ there exists $c\in C$ with $(c,x)\in U$. It can be shown that a uniform space is $\omega$-narrow if it has countable discrete cellularity. What about the converse?

>**Problem 2.** Does every $\omega$-narrow uniform space have countable discrete cellularity?

## Projects

Projects (3)

Lecture notes for introductory course
in the Classical Set Theory, based on
von Neumann-Bernays-Godel Axioms.

We explore the Borel complexity of some basic families of subsets of a countable group (large,
small, thin, sparse and other) dened by the size of their elements. Applying the obtained results to the
Stone-Cech compactication G of G, we prove, in particular, that the closure of the minimal ideal of G is
of type F.