Mathematics - General Topology Publications (50)

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Mathematics - General Topology Publications

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


We find (completeness type) conditions on topological semilattices $X,Y$ guaranteeing that each continuous homomorphism $h:X\to Y$ has closed image $h(X)$ in $Y$. Read More


A set $A$ of integers is called total if there is an algorithm which, given an enumeration of $A$, enumerates the complement of $A$, and called cototal if there is an algorithm which, given an enumeration of the complement of $A$, enumerates $A$. Many variants of totality and cototality have been studied in computability theory. In this note, by an effective forcing construction with strongly infinite dimensional Cantor manifolds, which can be viewed as an effectivization of Zapletal's "half-Cohen" forcing (i. Read More


We show that the enveloping space $X_G$ of a partial action of a Polish group $G$ on a Polish space $X$ is a standard Borel space, that is to say, there is a topology $\tau$ on $X_G$ such that $(X_G, \tau)$ is Polish and the quotient Borel structure on $X_G$ is equal to $Borel(X_G,\tau)$. To prove this result we show a generalization of a theorem of Burgess about Borel selectors for the orbit equivalence relation induced by a group action and also show that some properties of the Vaught's transform are valid for partial actions of groups. Read More


Given cardinals ${\lambda}$ and ${\mu}$ we say that $[\mathbf B({\lambda})]^C$ is ${\mu}$-colorable if there is a coloring $f:\mathbf B({\lambda})\to{\mu}$ such that $f"Z={\mu}$ whenever a subspace $Z\subset \mathbf B({\lambda})$ is homeomorphic to the Cantor set, where $\mathbf B({\lambda})$ denotes the Baire space of weight ${\lambda}$. We prove that a crowded feebly compact regular space $X$ is ${\mu}$-resolvable provided $[\mathbf B({\lambda})]^C$ is ${\mu}$-colorable for each ${\lambda}<\hat c(X)$. Consequently, (a) every crowded pseudocompact space $X$ with $c(X)< (2^{\omega})^{+{\omega}}$ is $2^{\omega}$-resolvable; (b) if $V=L$, then every crowded pseudocompact space is $2^{\omega}$-resolvable. Read More


We say that a topological space X is selectively sequentially pseudocompact (SSP for short) if for every sequence (U_n) of non-empty open subsets of X, one can choose a point x_n in U_n for every n in such a way that the sequence (x_n) has a convergent subsequence. We show that the class of SSP spaces is closed under taking arbitrary products and continuous images, contains the class of all dyadic spaces and forms a proper subclass of the class of strongly pseudocompact spaces introduced recently by Garc\'ia-Ferreira and Ortiz-Castillo. We investigate basic properties of this new class and its relations with known compactness properties. Read More


We construct classifying spaces for discrete and compact Lie groups, with the property that they are topological groups and complete metric spaces in a natural way. We sketch a program in view of extending these constructions. Read More


Every crowded space $X$ is ${\omega}$-resolvable in the c.c.c generic extension $V^{Fn(|X|,2})$ of the ground model. Read More


We define a general framework that includes objects such as tilings, Delone sets, functions and measures, as a first step to discuss their almost periodicity in a systematic way. We define local derivability and mutual local derivability (MLD) between any two of these objects in order to describe their interrelation. We show several canonical maps in aperiodic order send an object P to a one that is MLD with P. Read More


A compact hyperbolic "cobweb" manifold (hyperbolic space form) of symbol $Cw(6,6,6)$ will be constructed in Fig.1,4,5 as a representant of a presumably infinite series $Cw(2p,2p,2p)$ $(3 \le p \in \bN$ natural numbers). This is a by-product of our investigations \cite{MSz16}. Read More


In this article we introduce a new type of Pascal pyramids. A regular squared mosaic in the hyperbolic plane yields a $(h^2r)$-cube mosaic in space $\mathbf{H}^2\!\times\!\mathbf{R}$ and the definition of the pyramid is based on this regular mosaic. The levels of the pyramid inherit some properties from the Euclidean and hyperbolic Pascal triangles. Read More


In 1938, Tarski proved that a formula is not intuitionistically valid if, and only if, it has a counter-model in the Heyting algebra of open sets of some topological space. In fact, Tarski showed that any Euclidean space R^n with n >= 1 suffices, as does e.g. Read More


For an orientation-preserving homeomorphism of the sphere, we prove that if a translation line does not accumulate in a fixed point, then it necessarily spirals towards a topological attractor. This is in analogy with the description of flow lines given by Poincar\'e-Bendixson theorem. We then apply this result to the study of invariant continua without fixed points, in particular to circloids and boundaries of simply connected open sets. Read More


We study computable topological spaces and semicomputable and computable sets in these spaces. In particular, we investigate conditions under which semicomputable sets are computable. We prove that a semicomputable compact manifold $M$ is computable if its boundary $\partial M$ is computable. Read More


Consider a surface $S$ and let $M\subset S$. If $S\setminus M$ is not connected, then we say $M$ \emph{separates} $S$, and we refer to $M$ as a \emph{separating set} of $S$. If $M$ separates $S$, and no proper subset of $M$ separates $S$, then we say $M$ is a \emph{minimal separating set} of $S$. Read More


Two channels are said to be equivalent if they are degraded from each other. The space of equivalent channels with input alphabet $X$ and output alphabet $Y$ can be naturally endowed with the quotient of the Euclidean topology by the equivalence relation. A topology on the space of equivalent channels with fixed input alphabet $X$ and arbitrary but finite output alphabet is said to be natural if and only if it induces the quotient topology on the subspaces of equivalent channels sharing the same output alphabet. Read More


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 a locally compact Hausdorff space has a one-point Hausdorff compactification if and only if it is non-compact. The one-point Hausdorff compactification is indeed obtained by adding the so called "point at infinity." Here we consider the analogous problem of existence of a one-point connectification, and keeping analogy, we prove that a locally connected normal space has a one-point normal connectification if and only if it has no compact component. 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