arXiv subjects

Taras Banakh

Publications and source records attributed to Taras Banakh.

At least 91 records · Page 5Linked to original sources

arXiv2019

Some Baire category properties of topological groups

We present several known and new results on the Baire category properties in topological groups. In particular, we prove that a Baire topological group $X$ is metrizable if and only if $X$ is point-cosmic if and only if $X$ is a $σ$-space. A topological group $X$ is Choquet if and only if its Raikov completion $\bar X$ is Choquet and $X$ is $G_δ$-dense in $\bar X$. A topological group $X$ is complete-metrizable if and only if $X$ is a point-cosmic Choquet space if and only if $X$ is a Choquet $σ$-space. Finally, we pose several open problem, in particular, if each Choquet topological group is strong Choquet.

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arXiv2019

$σ$-Continuous functions and related cardinal characteristics of the continuum

A function $f:X\to Y$ between topological spaces is called $σ$-$continuous$ (resp. $\barσ$-$continuous$) if there exists a (closed) cover $\{X_n\}_{n\inω}$ of $X$ such that for every $n\inω$ the restriction $f{\restriction}X_n$ is continuous. By $\mathfrak c_σ$ (resp. $\mathfrak c_{\barσ}$) we denote the largest cardinal $κ\le\mathfrak c$ such that every function $f:X\to\mathbb R$ defined on a subset $X\subset\mathbb R$ of cardinality $|X|<κ$ is $σ$-continuous (resp. $\barσ$-continuous). It is clear that $ω_1\le\mathfrak c_{\barσ}\le\mathfrak c_σ\le\mathfrak c$. We prove that $\mathfrak p\le\mathfrak q_0=\mathfrak c_{\barσ}=\min\{\mathfrak c_σ,\mathfrak b,\mathfrak q\}\le\mathfrak c_σ\le\min\{\mathrm{non}(\mathcal M),\mathrm{non}(\mathcal N)\}$. The equality $\mathfrak c_{\barσ}=\mathfrak q_0$ resolves a problem from the initial version of the paper.

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arXiv2019

On $κ$-bounded and $M$-compact reflections of topological spaces

For a topological space $X$ its reflection in a class $\mathsf T$ of topological spaces is a pair $(\mathsf T X,i_X)$ consisting of a space $\mathsf T X\in\mathsf T$ and continuous map $i_X:X\to \mathsf T X$ such that for any continuous map $f:X\to Y$ to a space $Y\in\mathsf T$ there exists a unique continuous map $\bar f:\mathsf T X\to Y$ such that $f=\bar f\circ i_X$. In this paper for an infinite cardinal $κ$ and a nonempty set $M$ of ultrafilters on $κ$, we study the reflections of topological spaces in the classes $\mathsf H_κ$ of $κ$-bounded Hausdorff spaces and $\mathsf H_M$ of $M$-compact Hausdorff spaces (a topological space $X$ is $κ$-bounded if the closures of subsets of cardinality $\leκ$ in $X$ are compact; $X$ is $M$-compact if any function $x:κ\to X$ has a $p$-limit in $M$ for every ultrafilter $p\in M$).

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arXiv2019

Lusin and Suslin properties of function spaces

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 $C_{{\downarrow}F}(X)$ 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 topology, respectively. For a metrizable space $X$ we prove the equivalence of the following statements: (1) $X$ is $σ$-compact, (2) $C_p(X)$ is Suslin, (3) $C_k(X)$ is Suslin, (4) $C_{{\downarrow}F}(X)$ is Suslin, (5) $C_p(X)$ is Lusin, (6) $C_k(X)$ is Lusin, (7) $C_{{\downarrow}F}(X)$ is Lusin, (8) $C_p(X)$ is $F_σ$-Lusin, (9) $C_k(X)$ is $F_σ$-Lusin, (10) $C_{{\downarrow}F}(X)$ is $C_{δσ}$-Lusin. Also we construct an example of a sequential $\aleph_0$-space $X$ with a unique non-isolated point such that the function spaces $C_p(X)$, $C_k(X)$ and $C_{{\downarrow}F}(X)$ are not Suslin.

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arXiv2019

The Lawson number of a semitopological semilattice

For a Hausdorff topologized semilattice $X$ its $Lawson\;\; number$ $\barΛ(X)$ is the smallest cardinal $κ$ such that for any distinct points $x,y\in X$ there exists a family $\mathcal U$ of closed neighborhoods of $x$ in $X$ such that $|\mathcal U|\leκ$ and $\bigcap\mathcal U$ is a subsemilattice of $X$ that does not contain $y$. It follows that $\barΛ(X)\le\barψ(X)$, where $\barψ(X)$ is the smallest cardinal $κ$ such that for any point $x\in X$ there exists a family $\mathcal U$ of closed neighborhoods of $x$ in $X$ such that $|\mathcal U|\leκ$ and $\bigcap\mathcal U=\{x\}$. We prove that a compact Hausdorff semitopological semilattice $X$ is Lawson (i.e., has a base of the topology consisting of subsemilattices) if and only if $\barΛ(X)=1$. Each Hausdorff topological semilattice $X$ has Lawson number $\barΛ(X)\leω$. On the other hand, for any infinite cardinal $λ$ we construct a Hausdorff zero-dimensional semitopological semilattice $X$ such that $|X|=λ$ and $\barΛ(X)=\barψ(X)=cf(λ)$. A topologized semilattice $X$ is called (i) $ω$-$Lawson$ if $\barΛ(X)\leω$; (ii) $complete$ if each non-empty chain $C\subset X$ has $\inf C\in\overline{C}$ and $\sup C\in\overline{C}$. We prove that for any complete subsemilattice $X$ of an $ω$-Lawson semitopological semilattice $Y$, the partial order $\le_X=\{(x,y)\in X\times X:xy=x\}$ 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 an $ω$-Lawson semitopological semilattice $Y$ the image $h(X)$ is closed in $Y$.

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arXiv2019

Minmax bornologies

A bornology $\mathcal{B}$ on a set $X$ is called minmax if the smallest and the largest coarse structures on $X$ compatible with $\mathcal{B}$ coincide. We prove that $\mathcal{B}$ is minmax if and only if the family $\mathcal B^\sharp=\{p\inβX:\{X\setminus B:B\in\mathcal B\}\subset p\}$ consists of ultrafilters which are pairwise non-isomorphic via $\mathcal B$-preserving bijections of $X$. Also we construct a minmax bornology $\mathcal B$ on $ω$ such that the set $\mathcal B^\sharp$ is infinite. We deduce this result from the existence of a closed infinite subset in $βω$ that consists of pairwise non-isomorphic ultrafilters.

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arXiv2019

Classifying homogeneous ultrametric spaces up to coarse equivalence

For every metric space $X$ we introduce two cardinal characteristics $\mathrm{cov}^\flat(X)$ and $\mathrm{cov}^\sharp(X)$ describing the capacity of balls in $X$. We prove that these cardinal characteristics are invariant under coarse equivalence and prove that two ultrametric spaces $X,Y$ are coarsely equivalent if $\mathrm{cov}^\flat(X)=\mathrm{cov}^\sharp(X)=\mathrm{cov}^\flat(Y)=\mathrm{cov}^\sharp(Y)$. This result implies that an ultrametric space $X$ is coarsely equivalent to an isometrically homogeneous ultrametric space if and only if $\mathrm{cov}^\flat(X)=\mathrm{cov}^\sharp(X)$. Moreover, two isometrically homogeneous ultrametric spaces $X,Y$ are coarsely equivalent if and only if $\mathrm{cov}^\sharp(X)=\mathrm{cov}^\sharp(Y)$ if and only if each of these spaces coarsely embeds into the other space. This means that the coarse structure of an isometrically homogeneous ultrametric space $X$ is completely determined by the value of the cardinal $\mathrm{cov}^\sharp(X)=\mathrm{cov}^\flat(X)$.

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arXiv2019

Haar-$\mathcal I$ sets: looking at small sets in Polish groups through compact glasses

Generalizing Christensen's notion of a Haar-null set and Darji's notion of a Haar-meager set, we introduce and study the notion of a Haar-$\mathcal I$ set in a Polish group. Here $\mathcal I$ is an ideal of subsets of some compact metrizable space $K$. A Borel subset $B\subset X$ of a Polish group $X$ is called Haar-$\mathcal I$ if there exists a continuous map $f:K\to X$ such that $f^{-1}(B+x)\in\mathcal I$ for all $x\in X$. Moreover, $B$ is generically Haar-$\mathcal I$ if the set of witness functions $\{f\in C(K,X):\forall x\in X\;\;f^{-1}(B+x)\in\mathcal I\}$ is comeager in the function space $C(K,X)$. We study (generically) Haar-$\mathcal I$ sets in Polish groups for many concrete and abstract ideals $\mathcal I$, and construct the corresponding distinguishing examples. We prove some results on Borel hull of Haar-$\mathcal I$ sets, generalizing results of Solecki, Elekes, Vidnyánszky, Doležal, Vlasǎk on Borel hulls of Haar-null and Haar-meager sets. Also we establish various Steinhaus properties of the families of (generically) Haar-$\mathcal I$ sets in Polish groups for various ideals $\mathcal I$.

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arXiv2019

On topological classification of normed spaces endowed with the weak topology or the topology of compact convergence

In this paper the weak topology on a normed space is studied from the viewpoint of infinite-dimensional topology. Besides the weak topology on a normed space $X$ (coinciding with the topology of uniform convergence on finite subsets of the dual space $X^*$), we consider the topology $c$ of uniform convergence on compact subsets of $X^*$. It is known that this topology coincides with the weak topology on bounded subsets of $X$, but unlike to the latter has much better topological properties (e.g., is stratifiable). We prove that for normed spaces $X,Y$ with separable duals the spaces $(X,weak)$, $(Y,weak)$ are sequentially homeomorphic if and only if $\mathcal W(X)=\mathcal W(Y)$, where $\mathcal W(X)$ is the class of topological spaces homeomorphic to closed bounded subsets of $(X,weak)$. Moreover, if $X,Y$ are Banach spaces which are isomorphic to their hyperplanes and have separale duals, then the spaces $(X,weak)$ and $(Y,weak)$ are sequentially homeomorphic if and only of the spaces $(X,c)$ and $(Y,c)$ are homeomorphic. To prove this result, we show that for a normed space $X$ which is isomorphic to its hyperpane and has separable dual, the space $(X,c)$ (resp. $(X,weak)$) is (sequentially) homeomorphic to the product $B\times\mathbb R^\infty$ of the weak unit ball $B$ of $X$ and the linear space $\mathbb R^\infty$ with countable Hamel basis and the strongest linear topology.

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arXiv2019

Automorphism groups of superextensions of finite monogenic semigroups

A family $\mathcal L$ of subsets of a set $X$ is called linked if $A\cap B\ne\emptyset$ for any $A,B\in\mathcal L$. A linked family $\mathcal M$ of subsets of $X$ is maximal linked if $\mathcal M$ coincides with each linked family $\mathcal L$ on $X$ that contains $\mathcal M$. The superextension $λ(X)$ of $X$ 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 automorphisms of the superextensions of finite monogenic semigroups and characteristic ideals in such semigroups. In particular, we describe the automorphism groups of the superextensions of finite monogenic semigroups of cardinality $\leq 5$.

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arXiv2019

Embeddings into countably compact Hausdorff spaces

In this paper we consider the problem of characterization of topological spaces that embed into countably compact Hausdorff spaces. We study the separation axioms of subspaces of countably compact Hausdorff spaces and construct an example of a regular separable scattered topological space which cannot be embedded into an Urysohn countably compact topological space but embeds into a Hausdorff countably compact space.

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arXiv2019

Supercompact minus compact is super

According to a folklore characterization of supercompact spaces, a compact Hausdorff space is supercompact if and only if it has a binary closed $k$-network. This characterization suggests to call a topological space $super$ if it has a binary closed $k$-network $\mathcal N$. The binarity of $\mathcal N$ means that every linked subfamily of $\mathcal N$ is centered. Therefore, a Hausdorff space is supercompact if and only if it is super and compact. In this paper we prove that the class of super spaces contains all GO-spaces, all supercompact spaces, all metrizable spaces, and all collectionwise normal $\aleph$-spaces. Moreover, the class of super spaces is closed under taking Tychonoff products and discretely dense sets in Tychonoff products. The superness of metrizable spaces implies that each compact metrizable space is supercompact, which was first proved by Strok and Szymański (1975) and then reproved by Mills (1979), van Douwen (1981), and D{\c e}bski (1984).

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arXiv2019

Linearly continuous functions and $F_σ$-measurability

The linear continuity of a function defined on a vector space means that its restriction on every affine line is continuous. For functions defined on $\mathbb R^m$ this notion is near to the separate continuity for which it is required only the continuity on the straight lines which are parallel to coordinate axes. The classical Lebesgue theorem states that every separately continuous function $f:\mathbb R^m\to\mathbb R$ is of the $(m-1)$-th Baire class. In this paper we prove that every linearly continuous function $f:\mathbb R^m\to\mathbb R$ is of the first Baire class. Moreover, we obtain the following result. If $X$ is a Baire cosmic topological vector space, $Y$ is a Tychonoff topological space and $f:X\to Y$ is a Borel-measurable (even BP-measurable) linearly continuous function, then $f$ is $F_σ$-measurable. Using this theorem we characterize the discontinuity point set of an arbitrary linearly continuous function on $\mathbb R^m$. In the final part of the article we prove that any $F_σ$-measurable function $f:\partial U\to \mathbb R$ defined on the boundary of a strictly convex open set $U\subset\mathbb R^m$ can be extended to a linearly continuous function $\bar f:X\to \mathbb R$. This fact shows that in the ``descriptive sense'' the linear continuity is not better than the $F_σ$-measurability.

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arXiv2019

Selection properties of the split interval and the Continuum Hypothesis

We prove that every usco multimap $Φ:X\to Y$ from a metrizable separable space $X$ to a GO-space $Y$ has an $F_σ$-measurable selection. On the other hand, for the split interval $\ddot{\mathbb I}$ and the projection $P:\ddot{\mathbb I}^2\to{\mathbb I}^2$ of its square onto the unit square ${\mathbb I}^2$, the usco multimap $P^{-1}:{\mathbb I}^2\multimap\ddot{\mathbb I}^2$ has a Borel ($F_σ$-measurable) selection if and only if the Continuum Hypothesis holds. This CH-example shows that know results on Borel selections of usco maps into fragmentable compact spaces cannot be extended to a wider class of compact spaces.

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arXiv2019

Zero-sum subsets of decomposable sets in Abelian groups

A subset $D$ of an Abelian group is $decomposable$ if $\emptyset\ne D\subset D+D$. In the paper we give partial answer to an open problem asking whether every finite decomposable subset $D$ of an Abelian group contains a non-empty subset $Z\subset D$ with $\sum Z=0$. For every $n\in\mathbb N$ we present a decomposable subset $D$ of cardinality $|D|=n$ in the cyclic group of order $2^n-1$ such that $\sum D=0$, but $\sum T\ne 0$ for any proper non-empty subset $T\subset D$. On the other hand, we prove that every decomposable subset $D\subset\mathbb R$ of cardinality $|D|\le 7$ contains a non-empty subset $Z\subset D$ of cardinality $|Z|\le\frac12|D|$ with $\sum Z=0$. For every $n\in\mathbb N$ we present a subset $D\subset\mathbb Z$ of cardinality $|D|=2n$ such that $\sum Z=0$ for some subset $Z\subset D$ of cardinality $|Z|=n$ and $\sum T\ne 0$ for any non-empty subset $T\subset D$ of cardinality $|T|<n=\frac12|D|$. Also we prove that every finite decomposable subset $D$ of an Abelian group contains two non-empty subsets $A,B$ such that $\sum A+\sum B=0$.

math.GR↗

arXiv2019

Scattered compact sets in continuous images of Čech-complete spaces

Assume hat a functionally Hausdorff space $X$ is a continuous image of a Čech complete space $P$ with Lindelöf number $l(P)<\mathfrak c$. Then the following conditions are equivalent: (i) every compact subset of $X$ is scattered, (ii) for every continuous map $f:X\to Y$ to a functionally Hausdorff space $Y$ the image $f(X)$ has cardinality $|f(X)|\le \max\{l(P),ψ(Y)\}$, (iii) no continuous map $f:X\to[0,1]$ is surjective. Also we prove the equivalence of the conditions: (a) $ω_1<\mathfrak b$, (b) a K-analytic space $X$ (with a unique non-isolated point) is countable if and only if every compact subset of $X$ is countable.

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arXiv2019

Baire category properties of function spaces with the Fell hypograph topology

For a Tychonoff space $X$ and a subspace $Y\subset\mathbb R$, we study Baire category properties of the space $C_{\downarrow F}(X,Y)$ of continuous functions from $X$ to $Y$, endowed with the Fell hypograph topology. We characterize pairs $X,Y$ for which the function space $C_{\downarrow F}(X,Y)$ is $\infty$-meager, meager, Baire, Choquet, strong Choquet, (almost) complete-metrizable or (almost) Polish.

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arXiv2019

On Baire category properties of function spaces $C_k'(X,Y)$

We prove that for a stratifiable scattered space $X$ of finite scattered height, the function space $C_k(X)$ endowed with the compact-open topology is Baire if and only if $X$ has the Moving Off Property of Gruenhage and Ma. As a byproduct of the proof we establish many interesting Baire category properties of the function spaces $C_k'(X,Y)=\{f\in C_k(X,Y):f(X')\subset\{*_Y\}\}$, where $X$ is a topological space, $X'$ is the set of non-isolated points of $X$, and $Y$ is a topological space with a distinguished point $*_Y$.

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