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Taras Banakh

Publications and source records attributed to Taras Banakh.

At least 163 records · Page 9Linked to original sources
arXiv2015

On the sequential closure of the set of continuous functions in the space of separately continuous functions

For separable metrizable spaces $X,Y$ and a metrizable topological group $Z$ by $S(X\times Y,Z)$ we denote the space of all separately continuous functions $f:X\times Y\to Z$ endowed with the topology of layer-wise uniform convergence, generated by the subbase consisting of the sets $[K_X\times K_Y,U]=\{f\in S(X\times Y,Z):f(K_X\times K_Y)\subset U\}$, where $U$ is an open subset of $Z$ and $K_X\subset X$, $K_Y\subset Y$ are compact sets one of which is a singleton. We prove that every separately continuous function $f:X\times Y\to Z$ with zero-dimensional image $f(X\times Y)$ is a limit of a sequence of jointly continuous functions in the topology of layer-wise uniform convergence.

math.GN↗
arXiv2015

Detecting $σZ_n$-sets in topological groups and linear metric spaces

We prove that if an analytic subset $A$ of a linear metric space $X$ is not contained in a $σZ_ω$-subset of $X$ then for every Polish convex set $K$ with dense affine hull in $X$ the sum $A+K$ is non-meager in $X$ and the sets $A+A+K$ and $A-A+K$ have non-empty interior in the completion $\bar X$ of $X$. This implies two results: (i) an analytic subgroup $A$ of a linear metric space $X$ is a $σZ_ω$-space if $A$ is not Polish and $A$ contains a Polish convex set $K$ with dense affine hull in $X$; (ii) a dense convex analytic subset $A$ of a linear metric space $X$ is a $σZ_ω$-space if $A$ contains no open Polish subspace and $A$ contains a Polish convex set $K$ with dense affine hull in $X$.

math.GN↗
arXiv2015

Cardinal invariants distinguishing permutation groups

We prove that for infinite cardinals $κ<λ$ the alternating group $Alt(λ)$ (of even permutations) of $λ$ is not embeddable into the symmetric group $Sym(κ)$ (of all permutations) of $κ$. To prove this fact we introduce and study several monotone cardinal group invariants which take value $κ$ on the groups $Alt(κ)$ and $Sym(κ)$.

math.GR↗
arXiv2015

Verbal covering properties of topological spaces

For any topological space $X$ we study the relation between the universal uniformity $\mathcal U_X$, the universal quasi-uniformity $q\mathcal U_X$ and the universal pre-uniformity $p\mathcal U_X$ on $X$. For a pre-uniformity $\mathcal U$ on a set $X$ and a word $v$ in the two-letter alphabet $\{+,-\}$ we define the verbal power $\mathcal U^v$ of $\mathcal U$ and study its boundedness numbers $\ell(\mathcal U^v)$ and $\bar \ell(\mathcal U^v)$. The boundedness numbers of the (Boolean operations over) the verbal powers of the canonical pre-uniformities $p\mathcal U_X$, $q\mathcal U_X$ and $\mathcal U_X$ yield new cardinal characteristics $\ell^v(X)$, $\bar \ell^v(X)$, $q\ell^v(X)$, $q\bar \ell^v(X)$, $u\ell(X)$ of a topological space $X$, which generalize all known cardinal topological invariants related to (star)-covering properties. We study the relation of the new cardinal invariants $\ell^v$, $\bar \ell^v$ to classical cardinal topological invariants such as Lindelöf number $\ell$, density $d$, and spread $s$. The simplest new verbal cardinal invariant is the foredensity $\ell^-(X)$ defined for a topological space $X$ as the smallest cardinal $κ$ such that for any neighborhood assignment $(O_x)_{x\in X}$ there is a subset $A\subset X$ of cardinality $|A|\leκ$ that meets each neighborhood $O_x$, $x\in X$. It is clear that $\ell^-(X)\le d(X)\le \ell^-(X)\cdot χ(X)$. We shall prove that $\ell^-(X)=d(X)$ if $|X|<\aleph_ω$. On the other hand, for every singular cardinal $κ$ (with $κ\le 2^{2^{cf(κ)}}$) we construct a (totally disconnected) $T_1$-space $X$ such that $\ell^-(X)=cf(κ)<κ=|X|=d(X)$.

math.GN↗
arXiv2015

Detecting topological and Banach fractals among zero-dimensional spaces

A topological space $X$ is called a topological fractal if $X=\bigcup_{f\in\mathcal F}f(X)$ for a finite system $\mathcal F$ of continuous self-maps of $X$, which is topologically contracting in the sense that for every open cover $\mathcal U$ of $X$ there is a number $n\in\mathbb N$ such that for any functions $f_1,\dots,f_n\in \mathcal F$, the set $f_1\circ\dots\circ f_n(X)$ is contained in some set $U\in\mathcal U$. If, in addition, all functions $f\in\mathcal F$ have Lipschitz constant $<1$ with respect to some metric generating the topology of $X$, then the space $X$ is called a Banach fractal. It is known that each topological fractal is compact and metrizable. We prove that a zero-dimensional compact metrizable space $X$ is a topological fractal if and only if $X$ is a Banach fractal if and only if $X$ is either uncountable or $X$ is countable and its scattered height $\hbar(X)$ is a successor ordinal. For countable compact spaces this classification was recently proved by M.Nowak.

math.GN↗
arXiv2015

Each regular paratopological group is completely regular

We prove that a semiregular topological space $X$ is completely regular if and only if its topology is generated by a normal quasi-uniformity. This characterization implies that each regular paratopological group is completely regular. This resolves an old problem in the theory of paratopological groups, which stood open for about 60 years. Also we define a natural uniformity on each paratopological group and using this uniformity prove that each (first countable) Hausdorff paratopological group is functionally Hausdorff (and submetrizable). This resolves another two known open problems in the theory of paratopological groups.

math.GN↗
arXiv2015

On the submetrizability number and $i$-weight of quasi-uniform spaces and paratopological groups

We derive many upper bounds on the submetrizability number and $i$-weight of paratopological groups and topological monoids with open shifts. In particular, we prove that each first countable Hausdorff paratopological group is submetrizable thus answering a problem of Arhangelskii posed in 2002. Also we construct an example of a zero-dimensional (and hence regular) Hausdorff paratopological abelian group $G$ with countable pseudocharacter which is not submetrizable. In fact, all results on the $i$-weight and submetrizability are derived from more general results concerning normally quasi-uniformizable and bi-quasi-uniformizable spaces.

math.GN↗
arXiv2014

The strong Pytkeev property in topological spaces

A topological space $X$ has the strong Pytkeev property at a point $x\in X$ if there exists a countable family $\mathcal N$ of subsets of $X$ such that for each neighborhood $O_x\subset X$ and subset $A\subset X$ accumulating at $x$, there is a set $N\in\mathcal N$ such that $N\subset O_x$ and $N\cap A$ is infinite. We prove that for any $\aleph_0$-space $X$ and any space $Y$ with the strong Pytkeev property at a point $y\in Y$ the function space $C_k(X,Y)$ has the strong Pytkeev property at the constant function $X\to \{y\}\subset Y$. If the space $Y$ is rectifiable, then the function space $C_k(X,Y)$ is rectifiable and has the strong Pytkeev property at each point. We also prove that for any pointed spaces $(X_n,*_n)$, $n\inω$, with the strong Pytkeev property their Tychonoff product and their small box-product both have the strong Pytkeev property at the distinguished point. We prove that a sequential rectifiable space $X$ has the strong Pytkeev property if and only if $X$ is metrizable or contains a clopen submetrizable $k_ω$-subspace. A locally precompact topological group is metrizable if and only if it contains a dense subgroup with the strong Pytkeev property.

math.GN↗
arXiv2014

Quasi-pseudometrics on quasi-uniform spaces and quasi-metrization of topological monoids

We define a notion of a rotund quasi-uniform space and describe a new direct construction of a (right-continuous) quasi-pseudometric on a (rotund) quasi-uniform space. This new construction allows to give alternative proofs of several classical metrizability theorems for (quasi-)uniform spaces and also obtain some new metrizability results. Applying this construction to topological monoids with open shifts, we prove that the topology of any (semiregular) topological monoid with open shifts is generated by a family of (right-continuous) left-subinvariant quasi-pseudometrics, which resolves an open problem posed by Ravsky in 2001. This implies that a topological monoid with open shifts is completely regular if and only if it is semiregular. Since each paratopological group is a topological monoid with open shifts these results apply also to paratopological groups.

math.GN↗
arXiv2014

The regularity of quotient paratopological groups

Let $H$ be a closed subgroup of a regular abelian paratopological group $G$. The group reflexion $G^\flat$ of $G$ is the group $G$ endowed with the strongest group topology, weaker that the original topology of $G$. We show that the quotient $G/H$ is Hausdorff (and regular) if $H$ is closed (and locally compact) in $G^\flat$. On the other hand, we construct an example of a regular abelian paratopological group $G$ containing a closed discrete subgroup $H$ such that the quotient $G/H$ is Hausdorff but not regular.

math.GR↗
arXiv2014

Large free sets in universal algebras

We prove that for each universal algebra $(A,\mathcal A)$ of cardinality $|A|\ge 2$ and an infinite set $X$ of cardinality $|X|\ge|\mathcal A|$, the $X$-th power $(A^X,\mathcal A^X)$ of the algebra $(A,\mathcal A)$ contains a free subset $\mathcal F\subset A^X$ of cardinality $|\mathcal F|=2^{|X|}$. This generalizes the classical Fichtenholtz-Kantorovitch-Hausdorff result on the existence of an independent family $\mathcal I\subset\mathcal P(X)$ of cardinality $|\mathcal I|=|\mathcal P(X)|$ in the Boolean algebra $\mathcal P(X)$ of subsets of an infinite set $X$.

math.LO↗
arXiv2014

Toehold Purchase Problem: A comparative analysis of two strategies

Toehold purchase, defined here as purchase of one share in a firm by an investor preparing a tender offer to acquire majority of shares in it, reduces by one the number of shares this investor needs for majority. In the paper we construct mathematical models for the toehold and no-toehold strategies and compare the expected profits of the investor and the probabilities of takeover the firm in both strategies. It turns out that the expected profits of the investor in both strategies coincide. On the other hand, the probability of takeover the firm using the toehold strategy is considerably higher comparing to the no-toehold strategy. In the analysis of the models we apply the apparatus of incomplete Beta functions and some refined bounds for central binomial coefficients.

q-fin.GN↗
arXiv2014

Topologically invariant $σ$-ideals on the Hilbert cube

We study and classify topologically invariant $σ$-ideals with a Borel base on the Hilbert cube and evaluate their cardinal characteristics. One of the results of this paper solves (positively) a known problem whether the minimal cardinalities of the families of Cantor sets covering the unit interval and the Hilbert cube are the same.

math.GT↗
arXiv2014

Classifying invariant $σ$-ideals with analytic base on good Cantor measure spaces

Let $X$ be a zero-dimensional compact metrizable space endowed with a strictly positive continuous Borel $σ$-additive measure $μ$ which is good in the sense that for any clopen subsets $U,V\subset X$ with $μ(U)<μ(V)$ there is a clopen set $W\subset V$ with $μ(W)=μ(U)$. We study $σ$-ideals with Borel base on $X$ which are invariant under the action of the group $H_μ(X)$ of measure-preserving homeomorphisms of $(X,μ)$, and show that any such $σ$-ideal $\mathcal I$ is equal to one of seven $σ$-ideals: $\{\emptyset\}$, $[X]^{\leω}$, $\mathcal E$, $\mathcal M\cap\mathcal N$, $\mathcal M$, $\mathcal N$, or $[X]^{\le \mathfrak c}$. Here $[X]^{\leκ}$ is the ideal consisting of subsets of cardiality $\leκ$ in $X$, $\mathcal M$ is the ideal of meager subsets of $X$, $\mathcal N=\{A\subset X:μ(A)=0\}$ is the ideal of null subsets of $(X,μ)$, and $\mathcal E$ is the $σ$-ideal generated by closed null subsets of $(X,μ)$.

math.GN↗
arXiv2014

Embedding topological fractals in universal spaces

Let $X$ be a universal (Urysohn) space. We prove that every topological fractal is homeomorphic (isometric) to the attractor $A_{\mathcal F}$ of a function system ${\mathcal F}$ on $X$ consisting of Rakotch contractions.

math.GN↗
arXiv2014

On critical cardinalities related to $Q$-sets

In this note we collect some known information and prove new results about the small uncountable cardinal $\mathfrak q_0$. The cardinal $\mathfrak q_0$ is defined as the smallest cardinality $|A|$ of a subset $A\subset \mathbb R$ which is not a $Q$-set (a subspace $A\subset\mathbb R$ is called a $Q$-set if each subset $B\subset A$ is of type $F_σ$ in $A$). We present a simple proof of a folklore fact that $\mathfrak p\le\mathfrak q_0\le\min\{\mathfrak b,\mathrm{non}(\mathcal N),\log(\mathfrak c^+)\}$, and also establish the consistency of a number of strict inequalities between the cardinal $\mathfrak q_0$ and other standard small uncountable cardinals. This is done by combining some known forcing results. A new result of the paper is the consistency of $\mathfrak{p} < \mathfrak{lr} < \mathfrak{q}_0$, where $\mathfrak{lr}$ denotes the linear refinement number. Another new result is the upper bound $\mathfrak q_0\le\mathrm{non}(\mathcal I)$ holding for any $\mathfrak q_0$-flexible cccc $σ$-ideal $\mathcal I$ on $\mathbb R$.

math.LO↗