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T. Banakh

Publications and source records attributed to T. Banakh.

At least 19 recordsLinked to original sources
arXiv2021

The $\ell^p$-metrization of functors with finite supports

Let $p\in[1,\infty]$ and $F:\mathbf{Set}\to\mathbf{Set}$ be a functor with finite supports in the category $\mathbf{Set}$ 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 $n\in\mathbb N$ and $a\in Fn$ the map $X^n\to FX$, $f\mapsto Ff(a)$, is non-expanding with respect to the $\ell^p$-metric $d^p_{X^n}$ on $X^n$. We prove that the distance $d^p_{FX}$ is a pseudometric if and only if the functor $F$ preserves singletons; $d^p_{FX}$ is a metric if $F$ preserves singletons and one of the following conditions holds: (1) the metric space $(X,d_X)$ is Lipschitz disconnected, (2) $p=1$, (3) the functor $F$ has finite degree, (4) $F$ preserves supports. We prove that for any Lipschitz map $f:(X,d_X)\to (Y,d_Y)$ between metric spaces the map $Ff:(FX,d^p_{FX})\to (FY,d^p_{FY})$ is Lipschitz with Lipschitz constant $\mathrm{Lip}(Ff)\le \mathrm{Lip}(f)$. If the functor $F$ is finitary, has finite degree (and preserves supports), then $F$ preserves uniformly continuous function, coarse functions, coarse equivalences, asymptotically Lipschitz functions, quasi-isometries (and continuous functions). For many dimension functions we prove the formula $\dim F^pX\le\mathrm{deg}(F)\cdot\dim X$. Using injective envelopes, we introduce a modification $\check d^p_{FX}$ of the distance $d^p_{FX}$ and prove that the functor $\check F^p:\mathbf{Dist}\to\mathbf{Dist}$, $\check F^p:(X,d_X)\mapsto (FX,\check d^p_{FX})$, in the category $\mathbf{Dist}$ of distance spaces preserves Lipschitz maps and isometries between metric spaces.

math.GN↗
arXiv2021

Embedding topological spaces into Hausdorff $κ$-bounded spaces

Let $κ$ be an infinite cardinal. A topological space $X$ is $κ$-bounded if the closure of any subset of cardinality $\leκ$ 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 sequentially compact separable regular space that cannot be embedded into a Hausdorff $ω$-bounded space.

math.GN↗
arXiv2021

Vietoris hyperspaces of scattered Priestley spaces

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 the topology, namely, it is a Priestley space. One of our main objectives is investigating Vietoris hyperspaces of general Priestley spaces, addressing the question when their topologies are Skula and computing the associated ordinal ranks. We apply our results to scattered compact spaces based on certain almost disjoint families, in particular, Lusin families and ladder systems.

math.GN↗
arXiv2018

Josefson-Nissenzweig property for $C_p$-spaces

The famous Rosenthal-Lacey theorem asserts that for each infinite compact space $K$ the Banach space $C(K)$ admits a quotient which is either a copy of $c_{0}$ or $\ell_{2}$. The aim of the paper is to study a natural variant of this result for the space $C_{p}(K)$ of continuous real-valued maps on $K$ with the pointwise topology. Following famous Josefson-Nissenzweig theorem for infinite-dimensional Banach spaces we introduce a corresponding property (called Josefson-Nissenzweig property, briefly, the JNP) for $C_{p}$-spaces. We prove: For a Tychonoff space $X$ the space $C_p(X)$ satisfies the JNP if and only if $C_p(X)$ has a quotient isomorphic to $c_{0}$ (with the product topology of $\mathbb R^\mathbb{N}$) if and only if $C_{p}(X)$ contains a complemented subspace, isomorphic to $c_0$. For a pseudocompact space $X$ the space $C_p(X)$ has the JNP if and only if $C_p(X)$ has a complemented metrizable infinite-dimensional subspace. This applies to show that for a Tychonoff space $X$ the space $C_p(X)$ has a complemented subspace isomorphic to $\mathbb R^{\mathbb N}$ or $c_0$ if and only if $X$ is not pseudocompact or $C_p(X)$ has the JNP. The space $C_{p}(β\mathbb{N})$ contains a subspace isomorphic to $c_0$ and admits a quotient isomorphic to $\ell_{\infty}$ but fails to have a quotient isomorphic to $c_{0}$. An example of a compact space $K$ without infinite convergent sequences with $C_{p}(K)$ containing a complemented subspace isomorphic to $c_{0}$ is constructed.

math.FA↗
arXiv2018

Metrizable quotients of $C_p$-spaces

The famous Rosenthal-Lacey theorem asserts that for each infinite compact set $K$ the Banach space $C(K)$ admits a quotient which is either a copy of $c$ or $\ell_{2}$. What is the case when the uniform topology of $C(K)$ is replaced by the pointwise topology? Is it true that $C_p(X)$ always has an infinite-dimensional separable (or better metrizable) quotient? In this paper we prove that for a Tychonoff space $X$ the function space $C_p(X)$ has an infinite-dimensional metrizable quotient if $X$ either contains an infinite discrete $C^*$-embedded subspace or else $X$ has a sequence $(K_n)_{n\in\mathbb N}$ of compact subsets such that for every $n$ the space $K_n$ contains two disjoint topological copies of $K_{n+1}$. Applying the latter result, we show that under $\lozenge$ there exists a zero-dimensional Efimov space $K$ whose function space $C_{p}(K)$ has an infinite-dimensional metrizable quotient. These two theorems essentially improve earlier results of Kąkol and Śliwa on infinite-dimensional separable quotients of $C_p$-spaces.

math.GN↗
arXiv2015

Kuratowski monoids of $n$-topological spaces

Generalizing the famous 14-set closure-complement Theorem of Kuratowski from 1922, we prove that for a set $X$ endowed with $n$ pairwise comparable topologies $τ_1\subset\dots\subsetτ_n$, by repeated application of the operations of complement and closure in the topologies $τ_1,\dots,τ_n$ to a subset $A\subset X$ we can obtains at most $2K(n)=2\sum_{i,j=0}^n\binom{i+j}{i}\binom{i+j}{j}$ distinct sets.

math.GN↗
arXiv2014

The $C_p$-stable closure of the class of separable metrizable spaces

Denote by $\mathbf C_p[\mathfrak M_0]$ the $C_p$-stable closure of the class $\mathfrak M_0$ of all separable metrizable spaces, i.e., $\mathbf C_p[\mathfrak M_0]$ is the smallest class of topological spaces that contains $\mathfrak M_0$ and is closed under taking subspaces, homeomorphic images, countable topological sums, countable Tychonoff products, and function spaces $C_p(X,Y)$. Using a recent deep result of Chernikov and Shelah (2014), we prove that $\mathbf C_p[\mathfrak M_0]$ coincides with the class of all Tychonoff spaces of cardinality strictly less than $\beth_{ω_1}$. Being motivated by the theory of Generalized Metric Spaces, we characterize also other natural $C_p$-type stable closures of the class $\mathfrak M_0$.

math.GN↗
arXiv2013

Topological structure of non-separable sigma-locally compact convex sets

For an infinite cardinal $κ$ let $\ell_2(κ)$ be the linear hull of the standard othonormal base of the Hilbert space $\ell_2(κ)$ of density $κ$. We prove that a non-separable convex subset $X$ of density $κ$ in a locally convex linear metric space if homeomorphic to the space (i) $\ell_2^f(κ)$ if and only if $X$ can be written as countable union of finite-dimensional locally compact subspaces, (ii) $[0,1]^ω\times \ell_2^f(κ)$ if and only if $X$ contains a topological copy of the Hilbert cube and $X$ can be written as a countable union of locally compact subspaces.

math.GT↗
arXiv2012

Algebraic and topological properties of some sets in $l_1$

For a sequence $x \in l_1 \setminus c_{00}$, one can consider the set $E(x)$ of all subsums of series $\sum_{n=1}^{\infty} x(n)$. Guthrie and Nymann proved that $E(x)$ is one of the following types of sets: (I) a finite union of closed intervals; (C) homeomorphic to the Cantor set; (MC) homeomorphic to the set $T$ of subsums of $\sum_{n=1}^\infty b(n)$ where $b(2n-1) = 3/4^n$ and $b(2n) = 2/4^n$. By $I$, $C$ and $MC$ we denote the sets of all sequences $x \in l_1 \setminus c_{00}$, such that $E(x)$ has the corresponding property. In this note we show that $I$ and $C$ are strongly $\mathfrak{c}$-algebrable and $MC$ is $\mathfrak{c}$-lineable. We show that $C$ is a dense $G_δ$-set in $l_1$ and $I$ is a true $F_σ$-set. Finally we show that $I$ is spaceable while $C$ is not spaceable.

math.GN↗
arXiv2012

Compactly convex sets in linear topological spaces

A convex subset X of a linear topological space is called compactly convex if there is a continuous compact-valued map $Φ:X\to exp(X)$ such that $[x,y]\subsetΦ(x)\cup Φ(y)$ for all $x,y\in X$. We prove that each convex subset of the plane is compactly convex. On the other hand, the space $R^3$ contains a convex set that is not compactly convex. Each compactly convex subset $X$ of a linear topological space $L$ has locally compact closure $\bar X$ which is metrizable if and only if each compact subset of $X$ is metrizable.

math.FA↗
arXiv2012

On closed embeddings of free topological algebras

Let $\mathcal K$ be a complete quasivariety of completely regular universal topological algebras of continuous signature $\mathcal E$ (which means that $\mathcal K$ is closed under taking subalgebras, Cartesian products, and includes all completely regular topological $\mathcal E$-algebras algebraically isomorphic to members of $\mathcal K$). For a topological space $X$ by $F(X)$ we denote the free universal $\mathcal E$-algebra over $X$ in the class $\mathcal K$. Using some extension properties of the Hartman-Mycielski construction we prove that for a closed subspace $X$ of a metrizable (more generally, stratifiable) space $Y$ the induced homomorphism $F(X)\to F(Y)$ between the respective free universal algebras is a closed topological embedding. This generalizes one result of V.Uspenskii concerning embeddings of free topological groups.

math.GN↗
arXiv2011

The dimension of the space of R-places of certain rational function fields

We prove that the space $M(K(x,y))$ of $\mathbb R$-places of the field $K(x,y)$ of rational functions of two variables with coefficients in a totally Archimedean field $K$ has covering and integral dimensions $\dim M(K(x,y))=\dim_\IZ M(K(x,y))=2$ and the cohomological dimension $\dim_G M(K(x,y))=1$ for any Abelian 2-divisible coefficient group $G$.

math.AG↗
arXiv2009

On metric spaces with the properties of de Groot and Nagata in dimension one

A metric space $(X,d)$ has the de Groot property $GP_n$ if for any points $x_0,x_1,...,x_{n+2}\in X$ there are positive indices $i,j,k\le n+2$ such that $i\ne j$ and $d(x_i,x_j)\le d(x_0,x_k)$. If, in addition, $k\in\{i,j\}$ then $X$ is said to have the Nagata property $NP_n$. It is known that a compact metrizable space $X$ has dimension $dim(X)\le n$ iff $X$ has an admissible $GP_n$-metric iff $X$ has an admissible $NP_n$-metric. We prove that an embedding $f:(0,1)\to X$ of the interval $(0,1)$ into a locally connected metric space $X$ with property $GP_1$ (resp. $NP_1$) is open provided $f$ is an isometric embedding (resp. $f$ has distortion $Dist(f)=\|f\|_\Lip\cdot\|f^{-1}\|_\Lip<2$). This implies that the Euclidean metric cannot be extended from the interval $[-1,1]$ to an admissible $GP_1$-metric on the triode $T=[-1,1]\cup[0,i]$. Another corollary says that a topologically homogeneous $GP_1$-space cannot contain an isometric copy of the interval $(0,1)$ and a topological copy of the triode $T$ simultaneously. Also we prove that a $GP_1$-metric space $X$ containing an isometric copy of each compact $NP_1$-metric space has density not less than continuum.

math.MG↗
arXiv2009

Connected economically metrizable spaces

A topological space is nonseparably connected if it is connected but all of its connected separable subspaces are singletons. We show that each connected sequential topological space X is the image of a nonseparably connected complete metric space Eco(X) under a monotone quotient map. The metric d of the space Eco(X) is economical in the sense that for each infinite subspace A of X the cardinality of the set {d(a,b):a,b in A} does not exceed the density of A. The construction of the space Eco(X) determines a functor Eco from the category Top of topological spaces and their continuous maps into the category Metr of metric spaces and their non-expanding maps.

math.GN↗
arXiv2009

The topology of systems of hyperspaces determined by dimension functions

Given a non-degenerate Peano continuum $X$, a dimension function $D:2^X_*\to[0,\infty]$ defined on the family $2^X_*$ of compact subsets of $X$, and a subset $Γ\subset[0,\infty)$, we recognize the topological structure of the system $(2^X,\D_{\leγ}(X))_{α\inΓ}$, where $2^X$ is the hyperspace of non-empty compact subsets of $X$ and $D_{\leγ}(X)$ is the subspace of $2^X$, consisting of non-empty compact subsets $K\subset X$ with $D(K)\leγ$.

math.GN↗