Mathematics - General Topology Publications (50)

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Mathematics - General Topology Publications

We present how to obtain non-comparable regular but not completely regular spaces. We analyze a generalization of Mysior's example, extracting its underlying purely set-theoretic framework. This enables us to build simple counterexamples, using the Niemytzki plane, the Songefrey plane or Lusin gaps. Read More


Let $X$ be a (data) set. Let $K(x,y)>0$ be a measure of the affinity between the data points $x$ and $y$. We prove that $K$ has the structure of a Newtonian potential $K(x,y)=\varphi(d(x,y))$ with $\varphi$ decreasing and $d$ a quasi-metric on $X$ under two mild conditions on $K$. Read More


In this paper, we continue to study pairwise ($k$-semi-)stratifiable bitopological spaces. Some new characterizations of pairwise $k$-semi-stratifiable bitopological spaces are provided. Relationships between pairwise stratifiable and pairwise $k$-semi-stratifiable bitopological spaces are further investigated, and an open question recently posed by Li and Lin in \cite{LL} is completely solved. Read More


In this paper, we study two classes of planar self-similar fractals $T_\varepsilon$ with a shifting parameter $\varepsilon$. The first one is a class of self-similar tiles by shifting $x$-coordinates of some digits. We give a detailed discussion on the disk-likeness ({\it i. Read More


We characterize which 3-dimensional Seifert manifolds admit transitive partially hyperbolic diffeomorphisms. In particular, a circle bundle over a higher-genus surface admits a transitive partially hyperbolic diffeomorphism if and only if it admits an Anosov flow. Read More


We introduce a new Vietoris-type hypertopology by means of the upper-Vietoris-type hypertopology defined by G. Dimov and D. Vakarelov [On Scott consequence systems, Fundamenta Informaticae, 33 (1998), 43-70] (it was called there {\em Tychonoff-type hypertopology}) and the lower-Vietoris-type hypertopology introduced by E. Read More


It is a classical theorem of Alexandroff that every locally compact non-compact Hausdorff space has a one-point Hausdorff compactification, obtained by adding a "point at infinity." Here we consider the analogous problem of existence of a one-point connectification. We prove that a space with no compact component has a one-point connectification. Read More


The subpower Higson corona of a proper metric space is defined in [J. Kucab, M. Zarichnyi, Subpower Higson corona of a metric space, Algebra and Discrete Mathematics 17(2014) n2, 280--287]. Read More


In analogy to the topological entropy for continuous endomorphisms of totally disconnected locally compact groups, we introduce a notion of topological entropy for continuous endomorphisms of locally linearly compact vector spaces. We study the fundamental properties of this entropy and we prove the Addition Theorem, showing that the topological entropy is additive with respect to short exact sequences. By means of Lefschetz Duality, we connect the topological entropy to the algebraic entropy in a Bridge Theorem. Read More


Margalit and Schleimer observed that Dehn twists on orientable surfaces have nontrivial roots. We study the problem of roots of Dehn twists t_c about nonseparating circles c in the mapping class group M(N_g) of a nonorientable surface N_g of genus g. We investigate the existence of roots and, following the work of McCullough, Rajeevsarathy and Monden, give a simple arithmetic description of their conjugacy classes. Read More


In this paper we investigate the properties of functional spaces with the help of selection principles. We also got answers to some open problems. Read More


The main purpose of this paper is to study $e$-separable spaces, originally introduced by Kurepa as $K_0'$ spaces; a space $X$ is $e$-separable iff $X$ has a dense set which is the union of countably many closed discrete sets. We primarily focus on the behaviour of $e$-separable spaces under products and the cardinal invariants that are naturally related to $e$-separable spaces. Our main results show that the statement "the product of at most $\mathfrak c$ many $e$-separable spaces is $e$-separable" depends on the existence of certain large cardinals and hence independent of ZFC. Read More


We prove that if $H$ is a topological group such that all closed subgroups of $H$ are separable, then the product $G\times H$ has the same property for every separable compact group $G$. Let $c$ be the cardinality of the continuum. Assuming $2^{\omega_1} = c$, we show that there exist: (1) pseudocompact topological abelian groups $G$ and $H$ such that all closed subgroups of $G$ and $H$ are separable, but the product $G\times H$ contains a closed non-separable $\sigma$-compact subgroup; (2) pseudocomplete locally convex vector spaces $K$ and $L$ such that all closed vector subspaces of $K$ and $L$ are separable, but the product $K\times L$ contains a closed non-separable $\sigma$-compact vector subspace. Read More


We answer a question of O. Kalenda and J. Spurn\'{y} and give an example of a completely regular hereditarily Baire space $X$ and a Baire-one function $f:X\to [0,1]$ which can not be extended to a Baire-one function on $\beta X$. Read More


We prove a weaker version of the Hilbert-Smith conjecture: If $G$ is a compact Hausdorff topological group acting freely and continuously on a topological manifold $M$, so that the orbit space $M/G$ is finite dimensional, then $G$ is a Lie group. We do so by showing that no continuous action of a compact zero-dimensional group on a finite-dimensional manifold, that yields a finite-dimensional orbit space, can be free. Along the way, using a general compact-Hausdorff-space version of the Borsuk-Ulam theorem, we show that, given two universal free actions of a compact zero-dimensional group $G$ on the Menger compacta $\mu^m$ and $\mu^n$ with $m>n$, there are no \mbox{$G$-equivariant} continuous maps $\mu^m\to \mu^n$. Read More


We construct a universal action of a countable locally finite group on a separable metric space by isometries. This single action contains all actions of all countable locally finite groups on all separable metric spaces as subactions. The main ingredient is an amalgamation of actions by isometries. Read More


We introduce a new class of $\varkappa$-metrizable spaces, namely countably $\varkappa$-metrizable spaces. We show that the class of all $\varkappa$-metrizable spaces is a proper subclass of counably $\varkappa$-metrizable spaces. On the other hand, for pseudocompact spaces the new class coincides with $\varkappa$-metrizable spaces. Read More


Given an abstract simplicial complex G, the connection graph G' of G has as vertex set the faces of the complex and connects two if they intersect. If A is the adjacency matrix of that connection graph, we prove that the Fredholm characteristic det(1+A) takes values in {-1,1} and is equal to the Fermi characteristic, which is the product of the w(x), where w(x)=(-1)^dim(x). The Fredholm characteristic is a special value of the Bowen-Lanford zeta function and has various combinatorial interpretations. Read More


We prove several theorems relating amenability of groups in various categories (discrete, definable, topological, automorphism group) to model-theoretic invariants (quotients by connected components, Lascar Galois group, G-compactness, ... Read More


2016Dec

This issue surveys some of the activities in the field since the previous issue, and announces a conference fully dedicated to the topic of selection principles. Read More


We introduce matrix algebra of subsets in metric spaces and we apply it to improve results of Yamauchi and Davila regarding Asymptotic Property C. Here is a representative result: Suppose $X$ is an $\infty$-pseudo-metric space and $n\ge 0$ is an integer. The asymptotic dimension of $X$ is at most $n$ if and only if for any real number $r > 0$ and any integer $m\ge 1$ there is an augmented $m\times (n+1)$-matrix $\mathcal{M}=[\mathcal{B} |\mathcal{A}]$ (that means $\mathcal{B}$ is a column-matrix and $\mathcal{A}$ is an $m\times n$-matrix) of subspaces of $X$ of scale-$r$-dimension $0$ such that $\mathcal{M}\cdot_\cap \mathcal{M}^T$ is bigger than or equal to the identity matrix and $B(\mathcal{A},r)\cdot_\cap B(\mathcal{A},r)^T$ is a diagonal matrix. Read More


A topological space $X$ is called almost discretely Lindel\"of if every discrete set $D \subset X$ is included in a Lindel\"of subspace of $X$. We say that the space $X$ is {\em $\mu$-sequential} if for every non-closed set $A \subset X$ there is a sequence of length $\le \mu$ in $A$ that converges to a point which is not in $A$. With the help of a technical theorem that involves elementary submodels, we establish the following two results concerning such spaces. Read More


For each commutative and integral quantale, making use of the fuzzy order between closed sets, a theory of sobriety for quantale-valued cotopological spaces is established based on irreducible closed sets. Read More


Two of the natural topologies for infinite graphs with edge-ends are Etop and Itop. In this paper, we study and characterize them. We show that Itop can be constructed by inverse limits of inverse systems of graphs with finitely many vertices. Read More


Known and new results on free Boolean topological groups are collected. An account of properties which these groups share with free or free Abelian topological groups and properties specific of free Boolean groups is given. Special emphasis is placed on the application of set-theoretic methods to the study of Boolean topological groups. Read More


We deal with the problem of reconstructing a finite topological space to within homeomorphism given (also to within homeomorphism) the quotient spaces obtained by identifying one point of the space with each one of the other points. Read More


Let D be a planar Jordan domain. We consider a pursuit-evasion, contamination clearing, or sensor sweep problem in which the sensors at each point in time are modeled by a continuous curve. Both time and space are continuous, and the intruders are invisible to the sensors. Read More


We introduce the algebraic entropy for continuous endomorphisms of locally linearly compact vector spaces over a discrete field, as the natural extension of the algebraic entropy for endomorphisms of discrete vector spaces. We show that the main properties of entropy continue to hold in the general context of locally linearly compact vector spaces, in particular we extend the Addition Theorem. Read More


We provide a very short elementary proof that every bounded separable metric group embeds into a monothetic bounded metric group, in such a way that the result of Morris and Pestov that every separable abelian topological group embeds into a monothetic group is an immediate corollary. We show that the boundedness assumption is essential. Read More


Given an iterated function system (IFS) of contractive similitudes, the theory of Gromov hyperbolic graph on the IFS has been established recently. In the paper, we introduce a notion of simple augmented tree which is a Gromov hyperbolic graph. By generalizing a combinatorial device of rearrangeable matrix, we show that there exists a near-isometry between the simple augmented tree and the symbolic space of the IFS, so that their hyperbolic boundaries are Lipschitz equivalent. Read More


We find sufficient conditions under which the product of spaces that have a $\pi$-tree also has a $\pi$-tree. These conditions give new examples of spaces with a $\pi$-tree: every at most countable power of the Sorgenfrey line and every at most countable power of the irrational Sorgenfrey line has a $\pi\!$-tree. Also we show that if a space has a $\pi$-tree, then its product with the Baire space, with the Sorgenfrey line, and with the countable power of the Sorgenfrey line also has a $\pi$-tree. Read More


There are different definitions of homological dimension of metric compacta involving either \v{C}ech homology or exact (Steenrod) homology. In this paper we investigate the relation between these homological dimensions with respect to different groups. It is shown that all homological dimensions of a metric compactum X with respect to any field coincide provided X is homologically locally connected with respect to the singular homology up to dimension n=dim X. Read More


Let $E(X,f)$ be the Ellis semigroup of a dynamical system $(X,f)$ where $X$ is a compact metric space. We analyze the cardinality of $E(X,f)$ for a compact countable metric space $X$. A characterization when $E(X,f)$ and $E(X,f)^* = E(X,f) \setminus \{ f^n : n \in \mathbb{N}\}$ are both finite is given. Read More


In this paper, we shall study categorial properties of the hyperspace of all nontrivial convergent sequences $\mathcal{S}_c(X)$ of a Fre\'ech-Urysohn space $X$, this hyperspace is equipped with the Vietoris topology. We mainly prove that $\mathcal{S}_c(X)$ is meager whenever $X$ is a crowded space, as a corollary we obtain that if $\mathcal{S}_c(X)$ is Baire, the $X$ has a dense subset of isolated points. As an interesting example $\mathcal{S}_c(\omega_1)$ has the Baire property, where $\omega_1$ carries the order topology (this answers a question from \cite{sal-yas}). Read More


The "square peg problem" or "inscribed square problem" of Toeplitz asks if every simple closed curve in the plane inscribes a (non-degenerate) square, in the sense that all four vertices of that square lie on the curve. By a variety of arguments of a "homological" nature, it is known that the answer to this question is positive if the curve is sufficiently regular. The regularity hypotheses are needed to rule out the possibility of arbitrarily small squares that are inscribed or almost inscribed on the curve; because of this, these arguments do not appear to be robust enough to handle arbitrarily rough curves. Read More


We prove that every almost discretely Lindel\"of first-countable Hausdorff space has cardinality at most continuum in ZFC, thus completely answering a question by Juh\'asz, Tkachuk and Wilson and a question implicitly asked by Juh\'asz, Szentmikl\'ossy and Soukup. As a byproduct of the methods of our proof we also obtain a partial answer to Arhangel'skii's old problem about the cardinality of Lindel\"of $T_1$ first-countable spaces. Using a different method we prove that under $2^{<\mathfrak{c}}=\mathfrak{c}$ (which is a consequence of Martin's Axiom, for example), every sequential almost discretely Lindel\"of Hausdorff space of pseudocharacter at most continuum has cardinality at most continuum. Read More


A topological space $X$ is defined to have an $\omega^\omega$-base if at each point $x\in X$ the space $X$ has a neighborhood base $(U_\alpha[x])_{\alpha\in\omega^\omega}$ such that $U_\beta[x]\subset U_\alpha[x]$ for all $\alpha\le\beta$ in $\omega^\omega$. We characterize topological and uniform spaces whose free (locally convex) topological vector spaces or free (Abelian or Boolean) topological groups have $\omega^\omega$-bases. Read More


In \cite{Chaber}, Chaber has proved that countably compact spaces with a quasi $G_{\delta }$-diagonal are compact. We prove that initially $\kappa $% -compact spaces with a quasi $G_{\kappa }$-diagonal are compact, for any infinite cardinal $\kappa . Read More


We are concerned with questions of the following type. Suppose that $G$ and $K$ are topological groups belonging to a certain class $\cal K$ of spaces, and suppose that $\phi:K \to G$ is an abstract (i.e. Read More


We show that it is consistent to have an uncountable sequential group of intermediate sequential order while no countable such groups exist. This is proved by adding $\omega_2$ Cohen reals to a model of $\diamondsuit$. Read More


A topological space $X$ is called strongly $\sigma$-metrizable if $X=\bigcup_{n\in\omega}X_n$ for an increasing sequence $(X_n)_{n\in\omega}$ of closed metrizable subspaces such that every convergence sequence in $X$ is contained in some $X_n$. If, in addition, every compact subset of $X$ is contained in some $X_n$, $n\in\omega$, then $X$ is called super $\sigma$-metrizable. Answering a question of V. Read More


We establish fixed point theorems for nonlinear contractions on a metric space (not essentially complete) endowed with an arbitrary binary relation. Our results extend, generalize, modify and unify several known results especially those contained in Samet and Turinici [Commun. Math. Read More


Let $H$ be the C*-algebra of a non-trivial compact quantum group acting freely on a unital C*-algebra $A$. Baum, D\k{a}browski and Hajac conjectured that there does not exist an equivariant $*$-homomorphism from $A$ to the equivariant noncommutative join C*-algebra $A\circledast^\delta H$. When $A$ is the C*-algebra of functions on a sphere, and $H$ is the C*-algebra of functions on ${\mathbb Z}/2{\mathbb Z}$ acting antipodally on the sphere, then the conjecture becomes the celebrated Borsuk-Ulam theorem. Read More


We characterize Ascoli spaces by showing that a Tychonoff space $X$ is Ascoli iff the canonical map from the free locally convex space $L(X)$ over $X$ into $C_k\big( C_k(X)\big)$ is an embedding of locally convex spaces. We prove that an uncountable direct sum of non-trivial locally convex spaces is not Ascoli, and a direct countable sum $E$ of metrizable locally convex spaces is Ascoli iff $E=\phi$. If $X=\underrightarrow{\lim} \, X_n$ is the inductive limit of a sequence $\{ X_n\}_{n\in\omega}$ of metrizable groups such that $X_n$ is closed in $X_{n+1}$ for every $n\in\omega$, then the following assertions are equivalent: (i) there is $m\in\omega$ such that $X_n$ is open in $X_{n+1}$ for every $n\geq m$ or all the $X_n$ are locally compact, (ii) $X$ is sequential; (iii) $X$ is an Ascoli space. Read More


We show that there are widely-connected spaces of arbitrarily large cardinality, answering a question by David P. Bellamy. Read More


Let E be a locally solid vector lattice. In this paper, we consider two particular vector subspaces of the space of all order bounded operators on E. With the aid of two appropriate topologies, we show that under some conditions, they establish both, locally solid vector lattices and topologically complete topological algebras. Read More


Let $X$ be a paracompact topological space and $Y$ be a Banach space. In this paper, we will characterize the Baire-1 functions $f:X\rightarrow{Y}$ by their graph: namely, we will show that $f$ is a Baire-1 function if and only if its graph $gr(f)$ is the intersection of a sequence $(G_n)_{n=1}^{\infty}$ of open sets in $X\times{Y}$ such that for all $x\in{X}$ and $n\in\mathbb{N}$ the vertical section of $G_n$ is a convex set, whose diameter tends to $0$ as $n\rightarrow\infty$. Afterwards, we will discuss a similar question concerning functions of higher Baire classes and formulate some generalized results in slightly different settings: for example we require the domain to be a metrized Suslin space, while the codomain is a separable Fr\'echet space. Read More