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Taras Banakh

Publications and source records attributed to Taras Banakh.

At least 127 records · Page 7Linked to original sources
arXiv2018

A parallel metrization theorem

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, we prove that for a cover $\mathcal C$ of a metrizable space $X$ the following conditions are equivalent: (i) the topology of $X$ is generated by a metric $d$ such that any two sets $A,B\in\mathcal C$ are parallel; (ii) the cover $\mathcal C$ is disjoint, lower semicontinuous and upper semicontinuous.

math.GN↗
arXiv2018

Haar-open sets: a right way of generalizing the Steinhaus sum theorem to non-locally compact groups

Let $X$ be the countable product of Abelian locally compact Polish groups and $A,B\subset X$ be two Borel sets, which are not Haar-null in $X$. We prove that the sum-set $A+B:=\{a+b:a\in A,\;\;b\in B\}$ is Haar-open in the sense that for any non-empty compact subset $K\subset X$ and point $p\in K$ there exists a point $x\in X$ such that the set $K\cap(A+B+x)$ is a neighborhood of $p$ in $K$. This is a generalization of the classical Steinhaus Theorem (1920) to non-locally compact groups. We do not know if this generalization holds for Banach spaces.

math.GN↗
arXiv2018

The continuity of additive and convex functions, which are upper bounded on non-flat continua in $\mathbb R^n$

We prove that for a continuum $K\subset \mathbb R^n$ the sum $K^{+n}$ of $n$ copies of $K$ has non-empty interior in $\mathbb R^n$ if and only if $K$ is not flat in the sense that the affine hull of $K$ coincides with $\mathbb R^n$. Moreover, if $K$ is locally connected and each non-empty open subset of $K$ in not flat, then for any (analytic) non-meager subset $A\subset K$ the sum $A^{+n}$ of $n$ copies of $A$ is not meager in $\mathbb R^n$ (and then the sum $A^{+2n}$ of $2n$ copies of the analytic set $A$ has non-empty interior in $\mathbb R^n$ and the set $(A-A)^{+n}$ is a neighborhood of zero in $\mathbb R^n$). This implies that a mid-convex function $f:D\to\mathbb R$, defined on an open convex subset $D\subset\mathbb R^n$ is continuous if it is upper bounded on some non-flat continuum in $D$ or on a non-meager analytic subset of a locally connected nowhere flat subset of $D$.

math.GN↗
arXiv2018

Realizing spaces as path-component spaces

The path component space of a topological space $X$ is the quotient space $π_0(X)$ 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 $Y\to π_0(Y)=X$ is a perfect map. Hence, many topological properties of $X$ transfer to $Y$. We apply this result to construct a compact space $X\subset \mathbb{R}^3$ for which the fundamental group $π_1(X,x_0)$ is an uncountable, cosmic, $k_ω$-topological group but for which the canonical homomorphism $ψ:π_1(X,x_0)\to \checkπ_1(X,x_0)$ to the first shape homotopy group is trivial.

math.GN↗
arXiv2018

Lower separation axioms via Borel and Baire algebras

Let $κ$ be an infinite regular cardinal. We define a topological space $X$ to be $T_{κ-Borel}$-space (resp. a $T_{κ-BP}$-space) if for every $x\in X$ the singleton $\{x\}$ belongs to the smallest $κ$-additive algebra of subsets of $X$ that contains all open sets (and all nowhere dense sets) in $X$. Each $T_1$-space is a $T_{κ-Borel}$-space and each $T_{κ-Borel}$-space is a $T_0$-space. On the other hand, $T_{κ-BP}$-spaces need not be $T_0$-spaces. We prove that a topological space $X$ is a $T_{κ-Borel}$-space (resp. a $T_{κ-BP}$-space) if and only if for each point $x\in X$ the singleton $\{x\}$ is the intersection of a closed set and a $G_{<κ}$-set in $X$ (resp. $\{x\}$ is either nowhere dense or a $G_{<κ}$-set in $X$). Also we present simple examples distinguishing the separation axioms $T_{κ-Borel}$ and $T_{κ-BP}$ for various infinite cardinals $κ$, and we relate the axioms to several known notions, which results in a quite regular two-dimensional diagram of lower separation axioms.

math.GN↗
arXiv2018

Automorphism groups of superextensions of groups

The superextension $λ(X)$ of a set $X$ consists of all maximal linked families on $X$. Any associative binary operation $*: X\times X \to X$ can be extended to an associative binary operation $*: λ(X)\timesλ(X)\toλ(X)$. In the paper we study isomorphisms of superextensions of groups and prove that two groups are isomorphic if and only if their superextensions are isomorphic. Also we describe the automorphism groups of superextensions of all groups of cardinality $\leq 5$.

math.GR↗
arXiv2017

Null-finite sets in metric groups and their applications

In the paper we introduce a new family of "small" sets which is tightly connected with two well known $σ$-ideals: of Haar-null sets and of Haar-meager sets. We define a subset $A$ of a topological group $X$ to be $\mathit{null}$-$\mathit{finite}$ if there exists an infinite compact subset $K\subset X$ such that for every $x\in X$ the intersection $K\cap (x+A)$ is finite. We prove that each null-finite Borel set in a complete metric Abelian group is Haar-null and Haar-meager. The Borel restriction in the above result is essential as each non-discrete metric Abelian group is the union of two null-finite sets. Applying null-finite sets to the theory of functional equations and inequalities, we prove that a mid-point convex function $f:G\to\mathbb R$ defined on an open convex subset $G$ of a metric linear space $X$ is continuous if it is upper bounded on a subset $B$ which is not null-finite and whose closure is contained in $G$. This gives an alternative short proof of a known generalization of Bernstein-Doetsch theorem (saying that a mid-point convex function $f:G\to\mathbb R$ defined on an open covex subset $G$ of a metric linear space $X$ is continuous if it is upper bounded on a non-empty open subset $B$ of $G$). Since Borel null-finite sets are Haar-meager and Haar-null, we conclude that a mid-point convex function $f:G\to\mathbb{R}$ defined on an open convex subset $G$ of a complete linear metric space $X$ is continuous if it is upper bounded on a Borel subset $B\subset G$ which is not Haar-null or not Haar-meager in $X$. The last result resolves an old problem in the theory of functional equations and inequalities posed by Baron and Ger in 1983.

math.GN↗
arXiv2017

A simple inductive proof of Levy-Steinitz theorem

We present a relatively simple inductive proof of the classical Levy-Steinitz Theorem saying that for a sequence $(x_n)_{n=1}^\infty$ in a finite-dimensional Banach space $X$ the set of all sums of rearranged series $\sum_{n=1}^\infty x_{σ(n)}$ is an affine subspace of $X$. This affine subspace is not empty if and only if for any linear functional $f:X\to \mathbb R$ the series $\sum_{n=1}^\infty f(x_{σ(n)})$ is convergent for some permutation $σ$ of $\mathbb N$. This gives an answer to a problem of Vaja Tarieladze, posed in Lviv Scottish Book in September, 2017. Also we construct a sequence $(x_n)_{n=1}^\infty$ in the torus $\mathbb T\times\mathbb T$ such that the series $\sum_{n=1}^\infty x_{σ(n)}$ is divergent for all permutations $σ$ of $\mathbb N$ but for any continuous homomorphism $f:\mathbb T^2\to\mathbb T$ to the circle group $\mathbb T:=\mathbb R/\mathbb Z$ the series $\sum_{n=1}^\infty f(x_{σ_f(n)})$ is convergent for some permutation $σ_f$ of $\mathbb N$. This example shows that the second part of Levy-Steinitz Theorem (characterizing sequences with non-empty set of potential sums) does not extend to locally compact Abelian groups.

math.FA↗
arXiv2017

Sequential rectifiable spaces of countable $cs^*$-character

We prove that each non-metrizable sequential rectifiable space $X$ of countable $cs^*$-character contains a clopen rectifiable submetrizable $k_ω$-subspace $H$ and admits an open disjoint cover by subspaces homeomorphic to clopen subspaces of $H$. This implies that each sequential rectifiable space with countable $cs^*$-character either is metrizable or else is a topological sum of submetrizable $k_ω$-spaces. Consequently, $X$ is submetrizable and paracompact. This answers a question of Lin and Shen posed in 2011.

math.GN↗
arXiv2017

Categorically closed topological groups

Let $\mathcal C$ be a subcategory of the category of topologized semigroups and their partial continuous homomorphisms. An object $X$ of the category ${\mathcal C}$ is called ${\mathcal C}$-closed if for each morphism $f:X\to Y$ of the category ${\mathcal C}$ the image $f(X)$ is closed in $Y$. In the paper we detect topological groups which are $\mathcal C$-closed for the categories $\mathcal C$ whose objects are Hausdorff topological (semi)groups and whose morphisms are isomorphic topological embeddings, injective continuous homomorphisms, continuous homomorphisms, or partial continuous homomorphisms with closed domain.

math.GN↗
arXiv2017

A quantitative generalization of Prodanov-Stoyanov Theorem on minimal Abelian topological groups

A topological group $X$ is defined to have $compact$ $exponent$ if for some number $n\in\mathbb N$ the set $\{x^n:x\in X\}$ has compact closure in $X$. Any such number $n$ will be called a compact exponent of $X$. Our principal result states that a complete Abelian topological group $X$ has compact exponent (equal to $n\in\mathbb N$) if and only if for any injective continuous homomorphism $f:X\to Y$ to a topological group $Y$ and every $y\in \bar{f(X)}$ there exists a positive number $k$ (equal to $n$) such that $y^k\in f(X)$. This result has many interesting implications: (1) an Abelian topological group is compact if and only if it is complete in each weaker Hausdorff group topology; (2) each minimal Abelian topological group is precompact (this is the famous Prodanov-Stoyanov Theorem); (3) a topological group $X$ is complete and has compact exponent if and only if it is closed in each Hausdorff paratopological group containing $X$ as a topoloical subgroup (this confirms an old conjecture of Banakh and Ravsky).

math.GN↗
arXiv2017

Quasicontinuous functions with values in Piotrowski spaces

A topological space $X$ is called Piotrowski if every quasicontinuous map $f:Z\to X$ from a Baire space $Z$ to $X$ has a continuity point. In this paper we survey known results on Piotrowski spaces and investigate the relation of Piotrowski spaces to strictly fragmentable, Stegall, and game determined spaces. Also we prove that a Piotrowski Tychonoff space $X$ contains a dense (completely) metrizable Baire subspace if and only if $X$ is Baire (Choquet).

math.GN↗
arXiv2017

Difference bases in finite Abelian groups

A subset $B$ of a group $G$ is called a difference basis of $G$ if each element $g\in G$ can be written as the difference $g=ab^{-1}$ of some elements $a,b\in B$. The smallest cardinality $|B|$ of a difference basis $B\subset G$ is called the difference size of $G$ and is denoted by $Δ[G]$. The fraction $ð[G]:=\frac{Δ[G]}{\sqrt{|G|}}$ is called the difference characteristic of $G$. Using properies of the Galois rings, we prove recursive upper bounds for the difference sizes and characteristics of finite Abelian groups. In particular, we prove that for a prime number $p\ge 11$, any finite Abelian $p$-group $G$ has difference characteristic $ð[G]<\frac{\sqrt{p}-1}{\sqrt{p}-3}\cdot\sup_{k\in\mathbb N}ð[C_{p^k}]<\sqrt{2}\cdot\frac{\sqrt{p}-1}{\sqrt{p}-3}$. Also we calculate the difference sizes of all Abelian groups of cardinality $<96$.

math.CO↗
arXiv2017

Difference bases in dihedral groups

A subset $B$ of a group $G$ is called a difference basis of $G$ if each element $g\in G$ can be written as the difference $g=ab^{-1}$ of some elements $a,b\in B$. The smallest cardinality $|B|$ of a difference basis $B\subset G$ is called the difference size of $G$ and is denoted by $Δ[G]$. The fraction $ð[G]:=Δ[G]/{\sqrt{|G|}}$ is called the difference characteristic of $G$. We prove that for every $n\in\mathbb N$ the dihedral group $D_{2n}$ of order $2n$ has the difference characteristic $\sqrt{2}\leð[D_{2n}]\leq\frac{48}{\sqrt{586}}\approx1.983$. Moreover, if $n\ge 2\cdot 10^{15}$, then $ð[D_{2n}]<\frac{4}{\sqrt{6}}\approx1.633$. Also we calculate the difference sizes and characteristics of all dihedral groups of cardinality $\le80$.

math.CO↗
arXiv2017

Difference bases in cyclic groups

A subset $B$ of an Abelian group $G$ is called a difference basis of $G$ if each element $g\in G$ can be written as the difference $g=a-b$ of some elements $a,b\in B$. The smallest cardinality $|B|$ of a difference basis $B\subset G$ is called the difference size of $G$ and is denoted by $Δ[G]$. We prove that for every $n\in\mathbb N$ the cyclic group $C_n$ of order $n$ has difference size $\frac{1+\sqrt{4|n|-3}}2\le Δ[C_n]\le\frac32\sqrt{n}$. If $n\ge 9$ (and $n\ge 2\cdot 10^{15}$), then $Δ[C_n]\le\frac{12}{\sqrt{73}}\sqrt{n}$ (and $Δ[C_n]<\frac2{\sqrt{3}}\sqrt{n}$). Also we calculate the difference sizes of all cyclic groups of cardinality $\le 100$.

math.CO↗
arXiv2017

Isometric copies of directed trees in orientations of graphs

For every $n\in\mathbb N$ we construct a finite graph $G$ such that every orientation $\vec G$ of $G$ contains an isometric copy of any oriented tree on $n$ vertices, and evaluate the smallest possible cardinality of $G$. On the other hand, we prove that every graph $G$ admits an orientation containing no directed $ω$-paths of infinite diameter.

math.CO↗
arXiv2017

Spaces of nonnegatively curved surfaces

We determine the homeomorphism type of the space of smooth complete nonnegatively curved metrics on surfaces of positive Euler characteristic equipped with the topology of $C^γ$ uniform convergence on compact sets, when $γ$ is infinite or is not an integer. If $γ=\infty$, the space of metrics is homeomorphic to the separable Hilbert space. If $γ$ is finite and not an integer, the space of metrics is homeomorphic to the countable power of the linear span of the Hilbert cube. We also prove similar results for some other spaces of metrics including the space of complete smooth Riemannian metrics on an arbitrary manifold.

math.DG↗