Mathematics - General Topology Publications (50)

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Mathematics - General Topology Publications

We show that the level sets of automorphisms of free groups with respect to the Lipschitz metric are connected as subsets of Culler-Vogtmann space. In fact we prove our result in a more general setting of deformation spaces. As applications, we give metric solutions of the conjugacy problem for irreducible automorphisms and the detection of reducibility. Read More


The Dold$-$Thom theorem states that for a sufficiently nice topological space, M, there is an isomorphism between the homotopy groups of the infinite symmetric product of M and the homology groups of M itself. The crux of most known proofs of this is to check that a certain map is a quasi-fibration. It is our goal to present a more direct proof of the Dold$-$Thom theorem which does appeal to any such fact. Read More


Here we define the notion of residually null sets, a $\sigma$-ideal of subsets of a Polish space which contains the meager sets and, in some contexts, generalizes the notion of universally null. From this $\sigma$-ideal we realize a $\sigma$-algebra of sets which is consistently a strict extension of the Baire property algebra. We then uncover a generalization to Pettis' Theorem which furnishes a new automatic continuity result. Read More


In this article, we use $\lambda$-sequences to derive common fixed points for a family of self-mappings defined on a complete $G$-metric space. We imitate some existing techniques in our proofs and show that the tools emlyed can be used at a larger scale. These results generalise well known results in the literature. Read More


An unanswered question in general topology asks whether the Stone-\v{C}ech compactification of a widely-connected space is necessarily an indecomposable continuum. We describe a property of $X$ that is necessary and sufficient in order for $\beta X$ to be indecomposable, and examine the case when $X$ is a separable metric space. We also construct two widely-connected sets in $\mathbb R ^3$ - one is contained in a composant of each of its compactifications, and the other fails to be indecomposable upon the addition of a single limit point. Read More


Let $\mathcal{P}$ be the class of combinatorial 3-dimensional simple polytopes $P$, different from a tetrahedron, without 3- and 4-belts of facets. By the results of Pogorelov and Andreev, a polytope $P$ admits a realisation in Lobachevsky space $\mathbb{L}^3$ with right dihedral angles if and only if $P \in \mathcal{P}$. We consider two families of smooth manifolds defined by regular 4-colourings of Pogorelov polytopes P: six-dimensional quasitoric manifolds over $P$ and three-dimensional small covers of $P$; the latter are also known as three-dimensional hyperbolic manifolds of Loebell type. Read More


We prove a Gauss-Bonnet formula X(G) = sum_x K(x), where K(x)=(-1)^dim(x) (1-X(S(x))) is a curvature of a vertex x with unit sphere S(x) in the Barycentric refinement G1 of a simplicial complex G. K(x) is dual to (-1)^dim(x) for which Gauss-Bonnet is the definition of Euler characteristic X. Because the connection Laplacian L'=1+A' of G is unimodular, where A' is the adjacency matrix of of the connection graph G', the Green function values g(x,y) = (1+A')^-1_xy are integers and 1-X(S(x))=g(x,x). Read More


We describe non-locally connected planar continua via the concepts of fiber and numerical scale. Given a continuum $X\subset\mathbb{C}$ and $x\in\partial X$, we show that the set of points $y\in \partial X$ that cannot be separated from $x$ by any finite set $C\subset \partial X$ is a continuum. This continuum is called the {\em modified fiber} $F_x^*$ of $X$ at $x$. Read More


The aim of this note is to characterize trees, endowed with coarse wedge topology, that have a retractional skeleton. We use this characterization to provide new examples of non-commutative Valdivia compact spaces that are not Valdivia. Read More


Representation theory of the quantum torus Hopf algebra, when the parameter $q$ is a root of unity, is studied. We investigate a decomposition map of the tensor product of two irreducibles into the direct sum of irreducibles, realized as a `multiplicity module' tensored with an irreducible representation. The isomorphism between the two possible decompositions of the triple tensor product yields a map ${\bf T}$ between the multiplicity modules, called the 6j-symbols. Read More


We consider a topological space with its subbase which induces a coding for each point. Every second-countable Hausdorff space has a subbase that is the union of countably many pairs of disjoint open subsets. A dyadic subbase is such a subbase with a fixed enumeration. Read More


In extension theory, in particular in dimension theory, it is frequently useful to represent a given compact metrizable space X as the limit of an inverse sequence of compact polyhedra. We are going to show that, for the purposes of extension theory, it is possible to replace such an X by a better metrizable compactum Z. This Z will come as the limit of an inverse sequence of triangulated polyhedra with simplicial bonding maps that factor in a certain way. Read More


It is a simple fact that a subgroup generated by a subset $A$ of an abelian group is the direct sum of the cyclic groups $\langle a\rangle$, $a\in A$ if and only if the set $A$ is independent. In [5] the concept of an $independent$ set in an abelian group was generalized to a $topologically$ $independent$ $set$ in a topological abelian group (these two notions coincide in discrete abelian groups). It was proved that a topological subgroup generated by a subset $A$ of an abelian topological group is the Tychonoff direct sum of the cyclic topological groups $\langle a\rangle$, $a\in A$ if and only if the set $A$ is topologically independent and absolutely Cauchy summable. Read More


We propose a definition of computable manifold by introducing computability as a structure that we impose to a given topological manifold, just in the same way as differentiability or piecewise linearity are defined for smooth and PL manifolds respectively. Using the framework of computable topology and Type-2 theory of effectivity, we develop computable versions of all the basic concepts needed to define manifolds, like computable atlases and (computably) compatible computable atlases. We prove that given a computable atlas $\Phi$ defined on a set $M$, we can construct a computable topological space $(M, \tau_\Phi, \beta_\Phi, \nu_\Phi)$, where $\tau_\Phi$ is the topology on $M$ induced by $\Phi$ and that the equivalence class of this computable space characterizes the computable structure determined by $\Phi$. Read More


We say that a subset $S$ of an infinite group $G$ is a Ramsey-product subset if, for any infinite subsets $X$, $Y$ of $G$, there exist $x \in X$ and $y\in Y$ such that $x y \in S$ and $ y x \in S$ . We show that the family $\varphi$ of all Ramsey-product subsets of $G$ is a filter and $\varphi$ defines the subsemigroup $ \overline{G^*G^*}$ of the semigroup $G^*$ of all free ultrafilters on $G$. Read More


Complex networks are graph representation of complex systems from the real-world. They are ubiquitous in biological, ecological, social and infrastructural systems. Here we study a transformation of these complex networks into simplicial complexes, where cliques represent the simplicies of the complex. Read More


We prove that hereditarily Lindel\"of space which is $F_{\sigma\delta}$ in some compactification is absolutely $F_{\sigma\delta}$. In particular, this implies that any separable Banach space is absolutely $F_{\sigma\delta}$ when equipped with the weak topology. Read More


The regular open subsets of a topological space form a Boolean algebra, where the `join' of two regular open sets is the interior of the closure of their union. A `credence' is a finitely additive probability measure on this Boolean algebra, or on one of its subalgebras. We develop a theory of integration for such credences. Read More


We extend the notion of localic completion of generalised metric spaces by Steven Vickers to the setting of generalised uniform spaces. A generalised uniform space (gus) is a set X equipped with a family of generalised metrics on X, where a generalised metric on X is a map from the product of X to the upper reals satisfying zero self-distance law and triangle inequality. For a symmetric generalised uniform space, the localic completion lifts its generalised uniform structure to a point-free generalised uniform structure. Read More


Recently $S_{b}$-metric spaces are introduced as the generalizations of metric and $S$-metric spaces. In this paper we investigate some basic properties of this new space. We prove the Banach's contraction principle and a generalization of this principle on a complete $S_{b}$ -metric space. Read More


We study several types of multivalued functions in digital topology. Read More


A lion and a man move continuously in a space $X$. The aim of the lion is to capture his prey while the man wants to escape forever. Which of them has a strategy? This question has been studied for different metric domains. Read More


We describe the structure of Hausdorff locally compact semitopological $0$-bisimple inverse $\omega$-semigroups with compact maximal subgroups. In particular, we show that a Hausdorff locally compact semitopological $0$-bisimple inverse $\omega$-semigroup with a compact maximal subgroup is either compact or topologically isomorphic to the topological sum of its $\mathscr{H}$-classes. We describe the structure of Hausdorff locally compact semitopological $0$-bisimple inverse $\omega$-semigroups with a monothetic maximal subgroups. Read More


We investigate the classical Alexandroff-Borsuk problem in the category of non-triangulable manifolds: Given an $n$-dimensional compact non-triangulable manifold $M^n$ and $\varepsilon > 0$, does there exist an $\varepsilon$-map of $M^n$ onto an $n$-dimensional finite polyhedron which induces a homotopy equivalence? Read More


We introduce and analyze the following general concept of recurrence. Let $G$ be a group and let $X$ be a G-space with the action $G\times X\longrightarrow X$, $(g,x)\longmapsto gx$. For a family $\mathfrak{F}$ of subset of $X$ and $A\in \mathfrak{F}$, we denote $\Delta_{\mathfrak{F}}(A)=\{g\in G: gB\subseteq A$ for some $B\in \mathfrak{F}, \ B\subseteq A\}$, and say that a subset $R$ of $G$ is $\mathfrak{F}$-recurrent if $R\bigcap \Delta_{\mathfrak{F}} (A)\neq\emptyset$ for each $A\in \mathfrak{F}$. Read More


We explore the Borel complexity of some basic families of subsets of a countable group (large, small, thin, sparse and other) defined by the size of their elements. Applying the obtained results to the Stone-\v{C}ech compactification $\beta G$ of $G$, we prove, in particular, that the closure of the minimal ideal of $\beta G$ is of type $F_{\sigma\delta}$. Read More


This paper contributes to the techniques of topo-algebraic recognition for languages beyond the regular setting as they relate to logic on words. In particular, we provide a general construction on recognisers corresponding to adding one layer of various kinds of quantifiers and prove a related Reutenauer-type theorem. Our main tools are codensity monads and duality theory. Read More


In this article, we present a new type of fixed point for single valed mapping in a $G$-complete $G$-metric space. Read More


In his seminal work \cite{pal:61}, R. Palais extended a substantial part of the theory of compact transformation groups to the case of proper actions of locally compact groups. Here we extend to proper actions some other important results well known for compact group actions. Read More


We note a generalization of Whyte's geometric solution to the von Neumann problem for locally compact groups in terms of Borel and clopen piecewise translations. This strengthens a result of Paterson on the existence of Borel paradoxical decompositions for non-amenable locally compact groups. Along the way, we study the connection between some geometric properties of coarse spaces and certain algebraic characteristics of their wobbling groups. Read More


A ballean (or coarse structure) is a set endowed with some family of subsets, the balls, is such a way that balleans with corresponding morphisms can be considered as asymptotic counterparts of uniform topological spaces. For a ballean $\mathcal{B}$ on a set $X$, the hyperballean $\mathcal{B}^{\flat}$ is a ballean naturally defined on the set $X^{\flat}$ of all bounded subsets of $X$. We describe all balleans with hyperballeans of bounded geometry and analyze the structure of these hyperballeans. Read More


Following [2], a Tychonoff space $X$ is Ascoli if every compact subset of $C_k(X)$ is equicontinuous. By the classical Ascoli theorem every $k$-space is Ascoli. We show that a strict $(LF)$-space $E$ is Ascoli iff $E$ is a Fr\'{e}chet space or $E=\phi$. Read More


Banach's fixed point theorem for contraction maps has been widely used to analyze the convergence of iterative methods in non-convex problems. It is a common experience, however, that iterative maps fail to be globally contracting under the natural metric in their domain, making the applicability of Banach's theorem limited. We explore how generally we can apply Banach's fixed point theorem to establish the convergence of iterative methods when pairing it with carefully designed metrics. Read More


In this note, we discuss some fixed point theorems for contractive self mappings defined on a $G$-metric spaces. More precisely, we give fised point theorems for mappings with a contractive iterate at a point. Read More


In this article, we discuss fixed point results for $(\varepsilon,\lambda)$-uniformly locally contractive self mapping defined on $\varepsilon$-chainable $G$-metric type spaces. In particular, we show that under some more general conditions, certain fixed point results already obtained in the literature remain true. Moreover, in the last sections of this paper, we make use of the newly introduced notion of $\lambda$-sequences to derive new results. Read More


We define a family of quantum invariants of closed oriented $3$-manifolds using spherical multi-fusion categories. The state sum nature of this invariant leads directly to $(2+1)$-dimensional topological quantum field theories ($\text{TQFT}$s), which generalize the Turaev-Viro-Barrett-Westbury ($\text{TVBW}$) $\text{TQFT}$s from spherical fusion categories. The invariant is given as a state sum over labeled triangulations, which is mostly parallel to, but richer than the $\text{TVBW}$ approach in that here the labels live not only on $1$-simplices but also on $0$-simplices. Read More


We show that every countable Borel equivalence relation structurable by $n$-dimensional contractible simplicial complexes embeds into one which is structurable by such complexes with the further property that each vertex belongs to at most $M_n := 2^{n-1}(n^2+3n+2)-2$ edges; this generalizes a result of Jackson-Kechris-Louveau in the case $n = 1$. The proof is based on that of a classical result of Whitehead on countable CW-complexes. Read More


We study two-dimensional Finsler metrics of constant flag curvature and show that such Finsler metrics that admit a Killing field can be written in a normal form that depends on two arbitrary functions of one variable. Furthermore, we find an approach to calculate these functions for spherically symmetric Finsler surfaces of constant flag curvature. In particular, we obtain the normal form of the Funk metric on the unit disk D^2. Read More


In this paper we state a problem on rigidity of powers and give a solution of this problem for m=2. Our statement of this problem is elementary enough and does not require any knowledge of algebraic topology. Actually, this problem is related to unitary circle actions, rigid Hirzebruch genera and Kosniowski's conjecture. Read More


In this paper we consider some properties of a space B(X) of Borel functions on a set of reals X, with pointwise topology, that are stronger than separability. Read More


New results on the Baire product problem are presented. It is shown that an arbitrary product of almost locally ccc Baire spaces is Baire; moreover, the product of a Baire space and a 1st countable space which is $\beta$-unfavorable in the strong Choquet game is Baire. Read More


We study feebly compact topologies $\tau$ on the semilattice $\left(\exp_n\lambda,\cap\right)$ such that $\left(\exp_n\lambda,\tau\right)$ is a semitopological semilattice and prove that for any shift-continuous $T_1$-topology $\tau$ on $\exp_n\lambda$ the following conditions are equivalent: $(i)$ $\tau$ is countably pracompact; $(ii)$ $\tau$ is feebly compact; $(iii)$ $\tau$ is $d$-feebly compact; $(iv)$ $\left(\exp_n\lambda,\tau\right)$ is an $H$-closed space. Read More


We will show that every planar Peano continuum whose fundamental group is isomorphic to the fundamental group of a one-dimensional Peano continuum is homotopy equivalent to a one-dimensional Peano continuum. This answers a question asked by Cannon and Conner and illustrates the rigidity of the fundamental group for planar continua. Read More


In this paper, we present an interesting application of Baire's category theorem. Read More


We characterize the finite dimensional asymmetric normed spaces which are right bounded and the relation of this property with the natural compactness properties of the unit ball, as compactness and strong compactness. In contrast with some results found in the existing literature, we show that not all right bounded asymmetric norms have compact closed balls. We also prove that there are finite dimensional asymmetric normed spaces that satisfy that the closed unit ball is compact, but not strongly compact, closing in this way an open question on the topology of finite dimensional asymmetric normed spaces. Read More


We study several properties of the completed group ring $\hat{\mathbb{Z}}[[t^{\hat{\mathbb{Z}}}]]$ and the completed Alexander modules of knots. Then we prove that the profinite completions of knot groups determine the Alexander polynomials. Read More


We consider 9 natural tightness conditions for topological spaces that are all variations on countable tightness and investigate the interrelationships between them. Several natural open problems are raised. Read More


A finite abstract simplicial complex G defines two finite simple graphs: the Barycentric refinement G1, connecting two simplices if one is a subset of the other and the connection graph G', connecting two simplices if they intersect. We prove that the Poincare-Hopf value i(x)=1-X(S(x)), where X is Euler characteristics and S(x) is the unit sphere of a vertex x in G1, agrees with the Green function value g(x,x),the diagonal element of the inverse of (1+A'), where A' is the adjacency matrix of G'. By unimodularity, det(1+A') is the product of parities (-1)^dim(x) of simplices in G, the Fredholm matrix 1+A' is in GL(n,Z), where n is the number of simplices in G. Read More


In this paper, the notion of $(L,M)$-fuzzy convex structures is introduced. It is a generalization of $L$-convex structures and $M$-fuzzifying convex structures. In our definition of $(L,M)$-fuzzy convex structures, each $L$-fuzzy subset can be regarded as an $L$-convex set to some degree. Read More


In the paper, we provide an effective method for the Lipschitz equivalence of two-branch Cantor sets and three-branch Cantor sets by studying the irreducibility of polynomials. We also find that any two Cantor sets are Lipschitz equivalent if and only if their contraction vectors are equivalent provided one of the contraction vectors is homogeneous. Read More