arXiv subjects
Publications and source records attributed to Taras Banakh.
A Banach space $X$ has the $Mazur$-$Ulam$ $property$ if any isometry from the unit sphere of $X$ onto the unit sphere of any other Banach space $Y$ extends to a linear isometry of the Banach spaces $X,Y$. A Banach space $X$ is called $smooth$ if the unit ball has a unique supporting functional at each point of the unit sphere. We prove that each non-smooth 2-dimensional Banach space has the Mazur-Ulam property.
We prove that every isometry between the unit spheres of 2-dimensional Banach spaces extends to a linear isometry of the Banach spaces. This resolves the famous Tingley's problem in the class of 2-dimensional Banach spaces.
We prove that any isometry between the unit spheres of $C^2$-smooth (more generally, absolutely smooth) smooth Banach spaces extends to a linear isometry of the Banach spaces. This answers the famous Tingley's problem in the class of absolutely smooth $2$-dimensional Banach spaces.
Let $M$ be an ANR space and $X$ be a homotopy dense subspace in $M$. Assume that $M$ admits a continuous binary operation $*:M\times M\to M$ such that for every $x,y\in M$ the inclusion $x*y\in X$ holds if and only if $x,y\in X$. Assume also that there exist continuous unary operations $u,v:M\to M$ such that $x=u(x)*v(x)$ for all $x\in M$. Given a $2^ω$-stable $\mathbf Π^0_2$-hereditary weakly $\mathbf Σ^0_2$-additive class of spaces $\mathcal C$, we prove that the pair $(M,X)$ is strongly $(\mathbf Π^0_1\cap\mathcal C,\mathcal C)$-universal if and only if for any compact space $K\in\mathcal C$, subspace $C\in\mathcal C$ of $K$ and nonempty open set $U\subseteq M$ there exists a continuous map $f:K\to U$ such that $f^{-1}[X]=C$. This characterization is applied to detecting strongly universal Lawson semilattices.
Given a $T_0$ paratopological group $G$ and a class $\mathcal C$ of continuous homomorphisms of paratopological groups, we define the $\mathcal C$-$semicompletion$ $\mathcal C[G)$ and $\mathcal C$-$completion$ $\mathcal C[G]$ of the group $G$ that contain $G$ as a dense subgroup, satisfy the $T_0$-separation axiom and have certain universality properties. For special classes $\mathcal C$, we present some necessary and sufficient conditions on $G$ in order that the (semi)completions $\mathcal C[G)$ and $\mathcal C[G]$ be Hausdorff. Also, we give an example of a Hausdorff paratopological abelian group $G$ whose $\mathcal C$-semicompletion $\mathcal C[G)$ fails to be a $T_1$-space, where $\mathcal C$ is the class of continuous homomorphisms of sequentially compact topological groups to paratopological groups. In particular, the group $G$ contains an $ω$-bounded sequentially compact subgroup $H$ such that $H$ is a topological group but its closure in $G$ fails to be a subgroup.
A subset $A$ of a semigroup $S$ is called a $chain$ ($antichain$) if $xy\in\{x,y\}$ ($xy\notin\{x,y\}$) for any (distinct) elements $x,y\in S$. A semigroup $S$ is called ($anti$)$chain$-$finite$ if $S$ contains no infinite (anti)chains. We prove that each antichain-finite semigroup $S$ is periodic and for every idempotent $e$ of $S$ the set $\sqrt[\infty]{e}=\{x\in S:\exists n\in\mathbb N\;\;(x^n=e)\}$ is finite. This property of antichain-finite semigroups is used to prove that a semigroup is finite if and only if it is chain-finite and antichain-finite. Also we present an example of an antichain-finite semilattice that is not a union of finitely many chains.
We prove that for any coarse spaces $X_1,\dots,X_n$ of asymptotic dimension $\ge 1$, the product $X=X_1\times\dots\times X_n$ has asymptotic dimension $\ge n$. Another result states that a finitary coare space $Z$ has $\mathrm{asdim}(Z)\ge n$ if $Z$ admits an almost free action of the group $\mathbb Z^n$. We deduce these results from the following combinatorial result (that generalized the the Hex Theorem of Gale): for any cover $\mathcal F$ of a discrete box $K=k_1\times \dots \times k_n$, either some set $F\in\mathcal F$ contains a chain connecting two opposite faces of $K$ or there exists a set $B\subset K$ of diameter $\le 1$ such that $|\{F\in\mathcal F:F\cap B\ne\emptyset\}|>n$.
We discuss various modifications of separability, precompactnmess and narrowness in topological groups and test those modifications in the permutation groups $S(X)$ and $S_{<ω}(X)$.
We obtain many results and solve some problems about feebly compact paratopological groups. We obtain necessary and sufficient conditions for such a group to be topological. One of them is the quasiregularity. We prove that each $2$-pseudocompact paratopological group is feebly compact and that each Hausdorff $σ$-compact feebly compact paratopological group is a compact topological group. Our particular attention concerns periodic and topologically periodic groups. We construct examples of various compact-like paratopological groups which are not topological groups, among them a $T_0$ sequentially compact group, a $T_1$ $2$-pseudocompact group, a functionally Hausdorff countably compact group (under the axiomatic assumption that there is an infinite torsion-free abelian countably compact topological group without non-trivial convergent sequences), and a functionally Hausdorff second countable group sequentially pracompact group. We investigate cone topologies of paratopological groups which provide a general tool to construct pathological examples, especially examples of compact-like paratopological groups with discontinuous inversion. We find a simple interplay between the algebraic properties of a basic cone subsemigroup $S$ of a group $G$ and compact-like properties of two basic semigroup topologies generated by $S$ on the group $G$. We prove that the product of a family of feebly compact paratopological groups is feebly compact, and that a paratopological group $G$ is feebly compact provided it has a feebly compact normal subgroup $H$ such that a quotient group $G/H$ is feebly compact.
A locally convex space (lcs) $E$ is said to have an $ω^ω$-base if $E$ has a neighborhood base $\{U_α:α\inω^ω\}$ at zero such that $U_β\subseteq U_α$ for all $α\leqβ$. The class of lcs with an $ω^ω$-base is large, among others contains all $(LM)$-spaces (hence $(LF)$-spaces), strong duals of distinguished Fréchet lcs (hence spaces of distributions $D'(Ω)$). A remarkable result of Cascales-Orihuela states that every compact set in a lcs with an $ω^ω$-base is metrizable. Our main result shows that every uncountable-dimensional lcs with an $ω^ω$-base contains an infinite-dimensional metrizable compact subset. On the other hand, the countable-dimensional space $φ$ endowed with the finest locally convex topology has an $ω^ω$-base but contains no infinite-dimensional compact subsets. It turns out that $φ$ is a unique infinite-dimensional locally convex space which is a $k_{\mathbb{R}}$-space containing no infinite-dimensional compact subsets. Applications to spaces $C_{p}(X)$ are provided.
The $Golomb$ $space$ (resp. the $Kirch$ $space$) is the set $\mathbb N$ of positive integers endowed with the topology generated by the base consisting of arithmetic progressions $a+b\mathbb N_0=\{a+bn:n\ge 0\}$ where $a\in\mathbb N$ and $b$ is a (square-free) number, coprime with $a$. It is known that the Golomb space (resp. the Kirch space) is connected (and locally connected). By a recent result of Banakh, Spirito and Turek, the Golomb space has trivial homeomorphism group and hence is topologically rigid. In this paper we prove the topological rigidity of the Kirch space.
A topological space $X$ is cometrizable if it admits a weaker metrizable topology such that each point $x\in X$ has a (not necessarily open) neighborhood base consisting of metrically closed sets. We study the relation of cometrizable spaces to other generalized metric spaces and prove that all $\mathsf{as}$-cosmic spaces are cometrizable. Also, we present an example of a regular countable space of weight $ω_1$, which is not cometrizable. Under $ω_1=\mathfrak c$ this space contains no infinite compact subsets and hence is $\mathsf{cs}$-cosmic. Under $ω_1<\mathfrak p$ this countable space is Fréchet-Urysohn and is not $\mathsf{cs}$-cosmic.
For a metric space $X$, let $\mathsf FX$ be the space of all nonempty finite subsets of $X$ endowed with the largest metric $d^1_{\mathsf FX}$ such that for every $n\in\mathbb N$ the map $X^n\to\mathsf FX$, $(x_1,\dots,x_n)\mapsto \{x_1,\dots,x_n\}$, is non-expanding with respect to the $\ell^1$-metric on $X^n$. We study the completion of the metric space $\mathsf F^1\!X=(\mathsf FX,d^1_{\mathsf FX})$ and prove that it coincides with the space $\mathsf Z^1\!X$ of nonempty compact subsets of $X$ that have zero length (defined with the help of graphs). We prove that each subset of zero length in a metric space has 1-dimensional Hausdorff measure zero. A subset $A$ of the real line has zero length if and only if its closure is compact and has Lebesgue measure zero. On the other hand, for every $n\ge 2$ the Euclidean space $\mathbb R^n$ contains a compact subset of 1-dimensional Hausdorff measure zero that fails to have zero length.
A Hausdorff topological space $X$ is called $\textit{superconnected}$ (resp. $\textit{coregular}$) if for any nonempty open sets $U_1,\dots U_n\subseteq X$, the intersection of their closures $\bar U_1\cap\dots\cap\bar U_n$ is not empty (resp. the complement $X\setminus (\bar U_1\cap\dots\cap\bar U_n)$ is a regular topological space). A canonical example of a coregular superconnected space is the projective space $\mathbb Q\mathsf P^\infty$ of the topological vector space $\mathbb Q^{<ω}=\{(x_n)_{n\inω}\in \mathbb Q^ω:|\{n\inω:x_n\ne 0\}|<ω\}$ over the field of rationals $\mathbb Q$. The space $\mathbb Q\mathsf P^\infty$ is the quotient space of $\mathbb Q^{<ω}\setminus\{0\}^ω$ by the equivalence relation $x\sim y$ iff $\mathbb Q{\cdot}x=\mathbb Q{\cdot}y$. We prove that every countable second-countable coregular space is homeomorphic to a subspace of $\mathbb Q\mathsf P^\infty$, and a topological space $X$ is homeomorphic to $\mathbb Q\mathsf P^\infty$ if and only if $X$ is countable, second-countable, and admits a decreasing sequence of closed sets $(X_n)_{n\inω}$ such that (i) $X_0=X$, $\bigcap_{n\inω}X_n=\emptyset$, (ii) for every $n\inω$ and a nonempty open set $U\subseteq X_n$ the closure $\bar U$ contains some set $X_m$, and (iii) for every $n\inω$ the complement $X\setminus X_n$ is a regular topological space. Using this topological characterization of $\mathbb Q\mathsf P^\infty$ we find topological copies of the space $\mathbb Q\mathsf P^\infty$ among quotient spaces, orbit spaces of group actions, and projective spaces of topological vector spaces over countable topological fields.
We prove that an injective map $f:X\to Y$ between connected metrizable spaces $X,Y$ is continuous if for every connected subset $C\subset X$ the image $f(C)$ is connected and one of the following conditions is satisfied: (1) $Y$ is a 1-manifold and $X$ is compact; (2) $Y$ is a 2-manifold and $X$ is a closed $n$-manifold of dimension $n\ge 2$; (3) $Y$ is a 3-manifold and $X$ is a simply-connected closed $n$-manifold of dimension $n\ge 3$. This gives a partial answer to a problem of Willie Wong, posed on Mathoverflow.
For any compact Hausdorff space $K$ we construct a canonical finitary coarse structure $\mathcal E_{X,K}$ on the set $X$ of isolated points of $K$. This construction has two properties: $\bullet$ If a finitary coarse space $(X,\mathcal E)$ is metrizable, then its coarse structure $\mathcal E$ coincides with the coarse structure $\mathcal E_{X,\bar X}$ generated by the Higson compactification $\bar X$ of $X$; $\bullet$ A compact Hausdorff space $K$ coincides with the Higson compactification of the coarse space $(X,\mathcal E_{X,K})$ if the set $X$ is dense in $K$ and the space $K$ is Frechet-Urysohn. This implies that a compact Hausdorff space $K$ is homeomorphic to the Higson corona of some finitary coarse space if one of the following conditions holds: (i) $K$ is perfectly normal; (ii) $K$ has weight $w(K)\leω_1$ and character $χ(K)<\mathfrak p$. Under CH every (zero-dimensional) compact Hausdorff space of weight $\leω_1$ is homeomorphic to the Higson (resp. binary) corona of some cellular finitary coarse space.
A topological group $X$ is called $duoseparable$ if there exists a countable set $S\subseteq X$ such that $SUS=X$ for any neighborhood $U\subseteq X$ of the unit. We construct a functor $F$ assigning to each (abelian) topological group $X$ a duoseparable (abelain-by-cyclic) topological group $FX$, containing an isomorphic copy of $X$. In fact, the functor $F$ is defined on the category of unital topologized magmas. Also we prove that each $σ$-compact locally compact abelian topological group embeds into a duoseparable locally compact abelian-by-countable topological group.
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 $\{a+bn:n\ge 0\}$ with coprime $a,b$. We prove that the Golomb space $\mathbb N_τ$ is topologically rigid in the sense that its homeomorphism group is trivial. This resolves a problem posed by the first author at Mathoverflow in 2017.