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Taras Banakh

Publications and source records attributed to Taras Banakh.

At least 109 records · Page 6Linked to original sources
arXiv2019

Returning functions with closed graph are continuous

A function $f:X\to \mathbb R$ defined on a topological space $X$ is called returning if for any point $x\in X$ there exists a positive real number $M_x$ such that for every path-connected subset $C_x\subset X$ containing the point $x$ and any $y\in C_x\setminus\{x\}$ there exists a point $z\in C_x\setminus\{x,y\}$ such that $|f(z)|\le \max\{M_x,|f(y)|\}$. A topological space $X$ is called path-inductive if a subset $U\subset X$ is open if and only if for any path $γ:[0,1]\to X$ the preimage $γ^{-1}(U)$ is open in $[0,1]$. The class of path-inductive spaces includes all first-countable locally path-connected spaces and all sequential locally contractible space. We prove that a function $f:X\to \mathbb R$ defined on a path-inductive space $X$ is continuous if and only of it is returning and has closed graph. This implies that a (weakly) Świ\c atkowski function $f:\mathbb R\to\mathbb R$ is continuous if and only if it has closed graph, which answers a problem of Maliszewski, inscibed to Lviv Scottish Book.

math.GN↗
arXiv2019

The closedness of complete subsemilattices in functionally Hausdorff semitopological semilattices

A topologized semilattice $X$ is complete if each non-empty chain $C\subset X$ has $\inf C\in\bar C$ and $\sup C\in\bar C$. It is proved that for any complete subsemilattice $X$ of a functionally Hausdorff semitopological semilattice $Y$ the partial order $P=\{(x,y)\in X\times X:xy=x\}$ of $X$ is closed in $Y\times Y$ and hence $X$ is closed in $Y$. This implies that for any continuous homomorphism $h:X\to Y$ from a compete topologized semilattice $X$ to a functionally Hausdorff semitopological semilattice $Y$ the image $h(X)$ is closed in $Y$. The functional Hausdorffness of $Y$ in these two results can be replaced by the weaker separation axiom $\vec T_{2δ}$, defined in this paper.

math.GN↗
arXiv2019

On some functional generalizations of the regularity of topological spaces

We introduce and study some generalizations of regular spaces, which were motivated by studying continuity properties of functions between (regular) topological spaces. In particular, we prove that a first-countable Hausdorff topological space is regular if and only if it does not contain a topological copy of the Gutik hedgehog.

math.GN↗
arXiv2019

Positive answers to Koch's problem in special cases

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 classes of topological monoids. Namely, we show that a locally compact monothetic topological monoid is a compact topological group if and only if $S$ is a submonoid of a quasitopological group if and only if $S$ has open shifts if and only if $S$ is non-viscous in the sense of Averbukh. The last condition means that any neighborhood $U$ of the identity $1$ of $S$ and for any element $a\in S$ there exists a neighborhood $V$ of $a$ such that any element $x\in S$ with $(xV\cup Vx)\cap V\ne\emptyset$ belongs to the neighborhood $U$ of 1.

math.GR↗
arXiv2019

A metrizable semitopological semilattice with non-closed partial order

We construct a metrizable semitopological semilattice $X$ whose partial order $P=\{(x,y)\in X\times X:xy=x\}$ is a non-closed dense subset of $X\times X$. 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 convergent sequences.

math.GN↗
arXiv2019

Approximating points of a Banach space by points of an operator image

Answering one problem that has its origins in quantum mechanics, we prove that for any sequence $(A_n)_{n\in\mathbb N}$ of convex nowhere dense sets in a Banach space $X$ and any sequence $(\varepsilon_n)_{n=1}^\infty$ of positive real numbers with $\lim_{n\to\infty}\varepsilon_n=0$, the set $A=\{x\in X:\forall n\in\mathbb N\;\exists a\in A_n\;\;\|x-a\|< \varepsilon_n\}$ is nowhere dense in $X$.

math.FA↗
arXiv2019

On the spread of topological groups containing subsets of the Sorgenfrey line

We prove that any topological group $G$ containing a subspace $X$ of the Sorgenfrey line has spread $s(G)\ge s(X\times X)$. Under OCA, each topological group containing an uncountable subspace of the Sorgenfrey line has uncountable spread. This implies that under OCA a cometrizable topological group $G$ is cosmic if and only if it has countable spread. On the other hand, under CH there exists a cometrizable Abelian topological group that has hereditarily Lindelöf countable power and contains an uncountable subspace of the Sorgenfrey line. This cometrizable topological group has countable spread but is not cosmic.

math.GN↗
arXiv2019

Banalytic spaces and characterization of Polish groups

A topological space is defined to be banalytic (resp. analytic) if it is the image of a Polish space under a Borel (resp. continuous) map. A regular topological space is analytic if and only if it is banalytic and cosmic. Each (regular) banalytic space has countable spread (and under PFA is hereditarily Lindelöf). Applying banalytic spaces to topological groups, we prove that for a Baire topological group $X$ the following conditions are equivalent: (1) $X$ is Polish, (2) $X$ is analytic, (3) $X$ is banalytic and cosmic, (4) $X$ is banalytic and has countable pseudocharacter. Under PFA the conditions (1)--(4) are equivalent to the banalycity of $X$. The conditions (1)--(3) remain equivalent for any Baire semitopological group.

math.GN↗
arXiv2019

Functional boundedness of balleans

We pose some open problems related to boundedness of real-valued functions on balleans and coarse spaces. Also we prove that the Bergman property of groups is a coarse invariant. A special attention is payed to balleans on groups.

math.GR↗
arXiv2019

Constructing balleans

A ballean is a set endowed with a coarse structure. We introduce and explore three constructions of balleans from a pregiven family of balleans: bornological products, bouquets and combs. We analyze the smallest and the largest coarse structures on a set $X$ compatible with a given bornology on $X$.

math.GN↗
arXiv2019

A universal Banach space with a $K$-unconditional basis

For a constant $K\geq 1$ let $\mathfrak{B}_K$ be the class of pairs $(X,(\mathbf e_n)_{n\inω})$ consisting of a Banach space $X$ and an unconditional Schauder basis $(\mathbf e_n)_{n\inω}$ for $X$, having the unconditional basic constant $K_u\leq K$. Such pairs are called $K$-based Banach spaces. A based Banach space $X$ is rational if the unit ball of any finite-dimensional subspace spanned by finitely many basic vectors is a polyhedron whose vertices have rational coordinates in the Schauder basis of $X$. Using the technique of Fraïssé theory, we construct a rational $K$-based Banach space $\big(\mathbb U_K,(\mathbf e_n)_{n\inω}\big)$ which is $\mathfrak{RI}_K$-universal in the sense that each basis preserving isometry $f:Λ\to\mathbb U_K$ defined on a based subspace $Λ$ of a finite-dimensional rational $K$-based Banach space $A$ extends to a basis preserving isometry $\bar f:A\to\mathbb U_K$ of the based Banach space $A$. We also prove that the $K$-based Banach space $\mathbb U_K$ is almost $\mathfrak{FI}_1$-universal in the sense that any base preserving $\varepsilon$-isometry $f:Λ\to\mathbb U_K$ defined on a based subspace $Λ$ of a finite-dimensional $1$-based Banach space $A$ extends to a base preserving $\varepsilon$-isometry $\bar f:A\to\mathbb U_K$ of the based Banach space $A$. On the other hand, we show that no almost $\mathfrak{FI}_K$-universal based Banach space exists for $K>1$. The Banach space $\mathbb U_K$ is isomorphic to the complementably universal Banach space for the class of Banach spaces with an unconditional Schauder basis, constructed by Pełczyński in 1969.

math.FA↗
arXiv2018

Embedding fractals in Banach, Hilbert or Euclidean spaces

By a metric fractal we understand a compact metric space $K$ endowed with a finite family $\mathcal F$ of contracting self-maps of $K$ such that $K=\bigcup_{f\in\mathcal F}f(K)$. If $K$ is a subset of a metric space $X$ and each $f\in\mathcal F$ extends to a contracting self-map of $X$, then we say that $(K,\mathcal F)$ is a fractal in $X$. We prove that each metric fractal $(K,\mathcal F)$ is $\bullet$ isometrically equivalent to a fractal in the Banach spaces $C[0,1]$ and $\ell_\infty$; $\bullet$ bi-Lipschitz equivalent to a fractal in the Banach space $c_0$; $\bullet$ isometrically equivalent to a fractal in the Hilbert space $\ell_2$ if $K$ is an ultrametric space. We prove that for a metric fractal $(K,\mathcal F)$ with the doubling property there exists $k\in\mathbb N$ such that the metric fractal $(K,\mathcal F^{\circ k})$ endowed with the fractal structure $\mathcal F^{\circ k}=\{f_1\circ\dots\circ f_k:f_1,\dots,f_k\in\mathcal F\}$ is equi-Hölder equivalent to a fractal in a Euclidean space $\mathbb R^d$. This result is used to prove our main result saying that each finite-dimensional compact metrizable space $K$ containing an open uncountable zero-dimensional space $Z$ is homeomorphic to a fractal in a Euclidean space $\mathbb R^d$. For $Z$, being a copy of the Cantor set, this embedding result was proved by Duvall and Husch in 1992.

math.MG↗
arXiv2018

The normality and bounded growth of balleans

By a ballean we understand a set $X$ endowed with a family of entourages which is a base of some coarse structure on $X$. Given two unbounded ballean $X,Y$ with normal product $X\times Y$, we prove that the balleans $X,Y$ have bounded growth and the bornology of $X\times Y$ has a linearly ordered base. A ballean $(X,\mathcal E_X)$ is defined to have bounded growth if there exists a function $G$ assigning to each point $x\in X$ a bounded subset $G[x]\subset X$ so that for any bounded set $B\subset X$ the union $\bigcup_{x\in B}G[x]$ is bounded and for any entourage $E\in\mathcal E_X$ there exists a bounded set $B\subset X$ such that $E[x]\subset G[x]$ for all $x\in X\setminus B$. We prove that the product $X\times Y$ of two balleans has bounded growth if and only if $X$ and $Y$ have bounded growth and the bornology of the product $X\times Y$ has a linearly ordered base. Also we prove that a ballean $X$ has bounded growth (and the bornology of $X$ has a linearly ordered base) if its symmetric square $[X]^{\le 2}$ is normal (and the ballean $X$ is not ultranormal). A ballean $X$ has bounded growth and its bornology has a linearly ordered base if for some $n\ge 3$ and some subgroup $G\subset S_n$ the $G$-symmetric $n$-th power $[X]^n_G$ of $X$ is normal. On the other hand, we prove that for any ultranormal discrete ballean $X$ and every $n\ge 2$ the power $X^n$ is not normal but the hypersymmetric power $[X]^{\le n}$ of $X$ is normal. Also we prove that the finitary ballean of a group is normal if and only if it has bounded growth if and only if the group is countable.

math.GN↗
arXiv2018

On continuous self-maps and homeomorphisms of the Golomb space

The Golomb space $\mathbb N_τ$ is the set $\mathbb N$ of positive integers endowed with the topology $τ$ generated by the base consisting of arithmetic progressions $\{a+bn\}_{n=0}^\infty$ with coprime $a,b$. We prove that the Golomb space $\mathbb N_τ$ has continuum many continuous self-maps, contains a countable disjoint family of infinite closed connected subsets, the set $Π$ of prime numbers is a dense metrizable subspace of $\mathbb N_τ$, and each homeomorphism $h$ of $\mathbb N_τ$ has the following properties: $h(1)=1$, $h(Π)=Π$ and $Π_{h(x)}=h(Π_x)$ for all $x\in\mathbb N$. Here by $Π_x$ we denote the set of prime divisors of $x$.

math.GN↗
arXiv2018

Minimal covers of hypergraphs

For a hypergraph $H=(V,\mathcal E)$, a subfamily $\mathcal C\subseteq \mathcal E$ is called a cover of the hypergraph if $\bigcup\mathcal C=\bigcup\mathcal E$. A cover $\mathcal C$ is called minimal if each cover $\mathcal D\subseteq\mathcal C$ of the hypergraph $H$ coincides with $\mathcal C$. We prove that for a hypergraph $H$ the following conditions are equivalent: (i) each countable subhypergraph of $H$ has a minimal cover; (ii) each non-empty subhypergraph of $H$ has a maximal edge; (iii) $H$ contains no isomorphic copy of the hypergraph $(ω,ω)$. This characterization implies that a countable hypergraph $(V,\mathcal E)$ has a minimal cover if every infinite set $I\subseteq V$ contains a finite subset $F\subseteq I$ such that the family of edges $\mathcal E_F:=\{E\in\mathcal E:F\subseteq E\}$ is finite. Also we prove that a hypergraph $(V,\mathcal E)$ has a minimal cover if $\sup\{|E|:E\in\mathcal E\}<ω$ or for every $v\in V$ the family $\mathcal E_v:=\{E\in\mathcal E:v\in E\}$ is finite.

math.CO↗