Mathematics - General Topology Publications (50)

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Mathematics - General Topology Publications

We introduce a formal definition of a pattern poset which encompasses several previously studied posets in the literature. Using this definition we present some general results on the M\"obius function and topology of such pattern posets. We prove our results using a poset fibration based on the embeddings of the poset, where embeddings are representations of occurrences. Read More


In this work we define a primary spectrum of a commutative ring R with its Zariski topology $\mathfrak{T}$. We introduce several properties and examine some topological features of this concept. We also investigate differences between the prime spectrum and our primary spectrum. Read More


Let $X$ be a Hausdorff topological compact space with a partial order $% \preceq $ for which the order intervals are closed and let $\mathcal{F}$ be a nonempty commutative family of monotone maps from $X$ into $X$. In this paper, we apply a generalized version of the Knaster-Tarski theorem to show that if there exists $c\in X$ such that $c\preceq f(c)$ for every $f\in \mathcal{F}$, then the set of common fixed points of $\mathcal{F}$ is nonempty and has maximal and minimal elements. The result, specialized to the case of Banach spaces gives a general fixed point theorem that drops almost all assumptions from the recent results in this area. Read More


Let $\kappa$ be an infinite regular cardinal. We define a topological space $X$ to be $T_{\kappa-Borel}$-space (resp. a $T_{\kappa-BP}$-space) if for every $x\in X$ the singleton $\{x\}$ belongs to the smallest $\kappa$-additive algebra of subsets of $X$ that contains all open sets (and all nowhere dense sets) in $X$. Read More


We propose a generalization of continuous lattices and domains through the concept of enriched closure space, defined as a closure space equipped with a preclosure operator satisfying some compatibility conditions. In this framework we are able to define a notion of way-below relation; an appropriate definition of continuity then naturally follows. Characterizations of continuity of the enriched closure space and necessary and sufficient conditions for the interpolation property are proved. Read More


A well-known theorem of Nash-Williams shows that the collection of locally finite trees under the topological minor relation results in a BQO. Set theoretically, two very natural questions arise: (1) What is the number $\lambda$ of topological types of locally finite trees? (2) What are the possible sizes of an equivalence class of locally finite trees? For (1), clearly, $\omega \leq \lambda \leq \mathfrak{c}$ and Matthiesen refined it to $\omega_1 \leq \lambda \leq \mathfrak{c}$. Thus, this question becomes non-trivial when the Continuum Hypothesis is not assumed. Read More


In this paper, a new structure is defined on a topological space that equips the space with a concept of distance in order to do that firstly, a generalization of quasi-pseudo-metric space named R.O-metric space is introduced, and some of its basic properties is studied. Afterwards the concept of generalized R. Read More


We present a simple generalization of $W$-spaces introduced by G. Gruenhage. We show that this generalization leads to a strictly larger class of topological spaces which we call $\widetilde W$-spaces, and we provide several applications. Read More


The space of linearly recursive sequences of complex numbers admits two distinguished topologies. Namely, the adic topology induced by the ideal of those sequences whose first term is $0$ and the topology induced from the Krull topology on the space of complex power series via a suitable embedding. We show that these topologies are not equivalent. Read More


We provide characterizations of Lie groups as compact-like groups in which all closed zero-dimensional metric (compact) subgroups are discrete. The "compact-like" properties we consider include (local) compactness, (local) omega-boundedness, (local) countable compactness, (local) precompactness, (local) minimality and sequential completeness. Below is a sample of our characterizations: (i) A topological group is a Lie group if and only if it is locally compact and has no infinite compact metric zero-dimensional subgroups. Read More


In the paper we present various characterizations of chain-compact and chain-finite topological semilattices. A topological semilattice $X$ is called chain-compact (resp. chain-finite) if each closed chain in $X$ is compact (finite). Read More


The pinning down number $pd(X)$ of a topological space $X$ is the smallest cardinal $\kappa$ such that for every neighborhood assignment $\mathcal{U}$ on $X$ there is a set of size $\kappa$ that meets every member of $\mathcal{U}$. Clearly, $pd(X) \le d(X)$ and we call $X$ a pd-example if $pd(X) < d(X)$. We denote by $\mathbf{S}$ the class of all singular cardinals that are not strong limit. Read More


In order to understand the structure of the "typical" element of a homeomorphism group, one has to study how large the conjugacy classes of the group are. When typical means generic in the sense of Baire category, this is well understood, see e.g. Read More


We determine the homeomorphism type of the hyperspace of positively curved $C^\infty$ convex bodies in $\mathbb R^n$, and derive various properties of its quotient by the group of Euclidean isometries. We make a systematic study of hyperspaces of convex bodies that are at least $C^1$. We show how to destroy the symmetry of a family of convex bodies, and prove that this cannot be done modulo congruence. Read More


It is shown that CH implies the existence of a compact Hausdorff space that is countable dense homogeneous, crowded and does not contain topological copies of the Cantor set. This contrasts with a previous result by the author which says that for any crowded Hausdorff space $X$ of countable $\pi$-weight, if ${}^\omega{X}$ is countable dense homogeneous, then $X$ must contain a topological copy of the Cantor set. Read More


We provide a direct proof of a recent theorem of Ben-Yaacov, Melleray, and Tsankov. If $G$ is a Polish group and $X$ is a minimal, metrizable $G$-flow with all orbits meager, we use $X$ to produce a new $G$-flow $S_G(X)$ which is minimal and non-metrizable. Read More


Let ${\mathfrak C}$ be a monster model of an arbitrary theory $T$, $\bar \alpha$ any tuple of bounded length of elements of ${\mathfrak C}$, and $\bar c$ an enumeration of all elements of ${\mathfrak C}$. By $S_{\bar \alpha}({\mathfrak C})$ denote the compact space of all complete types over ${\mathfrak C}$ extending $tp(\bar \alpha/\emptyset)$, and $S_{\bar c}({\mathfrak C})$ is defined analogously. Then $S_{\bar \alpha}({\mathfrak C})$ and $S_{\bar c}({\mathfrak C})$ are naturally $Aut({\mathfrak C})$-flows. Read More


Let G be an abelian group. For a subset A of G, Cyc(A) denotes the set of all elements x of G such that the cyclic subgroup generated by x is contained in A, and G is said to have the small subgroup generating property (abbreviated to SSGP) if the smallest subgroup of G generated by Cyc(U) is dense in G for every neighbourhood U of zero of G. SSGP groups form a proper subclass of the class of minimally almost periodic groups. Read More


The existence of a countably compact group without non-trivial convergent sequences in ZFC alone is a major open problem in topological group theory. We give a ZFC example of a Boolean topological group G without non-trivial convergent sequences having the following "selective" compactness property: For each free ultrafilter p on N and every sequence {U_n:n in N} of non-empty open subsets of G one can choose a point x_n in U_n for all n in such a way that the resulting sequence {x_n:n in N} has a p-limit in G, that is, {n in N: x_n in V} belongs to p for every neighbourhood V of x in G. In particular, G is selectively pseudocompact (strongly pseudocompact) but not selectively sequentially pseudocompact. Read More


Given a metric space $(X,d)$, we say that a mapping $\chi: [X]^{2}\longrightarrow\{0.1\}$ is an isometric coloring if $d(x,y)=d(z,t)$ implies $\chi(\{x,y\})=\chi(\{z,t\})$. A free ultrafilter $\mathcal{U}$ on an infinite metric space $(X,d)$ is called metrically Ramsey if, for every isometric coloring $\chi$ of $[X]^{2}$, there is a member $U\in\mathcal{U}$ such that the set $[U]^{2}$ is $\chi$-monochrome. Read More


A continuum $K$ is a common model for the family ${\mathcal K}$ of continua if every member of ${\mathcal K}$ is a continuous image of $K$. We show that none of the following classes of spaces has a common model: 1) the class of strongly chaotic hereditarily indecomposable $n$-dimensional Cantor manifolds, for any given natural number $n$, 2) the class of strongly chaotic hereditarily indecomposable hereditarily strongly infinite-dimensional Cantor manifolds, 3) the class of strongly chaotic hereditarily indecomposable continua with transfinite dimension (small or large) equal to $\alpha$, for any given ordinal number $\alpha < \omega_{1}$. Read More


This article introduces proximal Cech complexes, restricted to the Euclidean plane. A Cechh complex of a set of points is the nerve of a collection of convex sets that are closed geometric balls all with the same radius, that have non-empty intersection. Both spatial and descriptive closed balls are considered. Read More


In this paper we have studied the idea of ideal completeness of function spaces Y to the power X with respect to pointwise uniformity and uniformity of uniform convergence. Further involving topological structure on X we have obtained relationships between the uniformity of uniform convergence on compacta on Y to the power X and uniformity of uniform convergence on Y to the power X in terms of I-Cauchy condition and I-convergence of a net. Also using the notion of a k-space we have given a sufficient condition for C(X,Y) to be ideal complete with respect to the uniformity of uniform convergence on compacta. Read More


For $X$ a metric space and $r>0$ a scale parameter, the Vietoris-Rips complex $VR_<(X;r)$ (resp. $VR_\leq(X;r)$) has $X$ as its vertex set, and a finite subset $\sigma\subseteq X$ as a simplex whenever the diameter of $\sigma$ is less than $r$ (resp. at most $r$). Read More


We present an internal characterization for the productively Lindel\"of property, thus answering a long-standing problem attributed to Tamano. We also present some results about the relation Alster spaces vs. productively Lindel\"of spaces. Read More


We extend to metrizable locally compact groups Rosenthal's theorem describing those Banach spaces containing no copy of `1. For that purpose, we transfer to general locally compact groups the notion of interpolation (I0) set, which was defined by Hartman and Ryll-Nardzewsky [25] for locally compact abelian groups. Thus we prove that for every sequence fgngnRead More


For a discrete group $G$, we use the natural correspondence between ideals in the Boolean algebra $ \mathcal{P}_G$ of subsets of $G$ and closed subsets in the Stone-$\check{C}$ech compactifi-cation $\beta G$ as a right topological semigroup to introduce and characterize some new ideals in $\beta G$. We show that if a group $G$ is either countable or Abelian then there are no closed ideals in $\beta G$ maximal in $G^*$, $G^* = \beta G \setminus G$, but this statement does not hold for the group $S_\kappa$ of all permutations of an infinite cardinal $\kappa$. We characterize the minimal closed ideal in $\beta G$ containing all idempotents of $G^*$. Read More


Let $\sR$ be an epireflective category of $\topo$ and let $F_\sR$\, be the epireflective functor associated with $\sR$. If $\sA$ denotes a (semi)topological algebraic subcategory of $\topo$, we study when $F_\sR\,(A)$ is an epireflective subcategory of $\sA$. We prove that this is always the case for semi-topological structures and we find some sufficient conditions for topological algebraic structures. Read More


We explore extensions of domain theoretic concepts, like completeness and continuity, replacing order relations with non-symmetric distances. Read More


In this work we deal with the preservation by $G_\delta$-refinements. We prove that for $\mathrm{SP}$-scattered spaces the metacompactness, paralindel\"ofness, metalindel\"ofness and linear lindel\"ofness are preserved by $G_\delta$-refinements. In this context we also consider some other generalizations of discrete spaces like $\omega$-scattered and $N$-scattered. Read More


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


We present different extensions of the Banach contraction principle in the $G$-metric space setting. More precisely, we consider mappings for which the contractive condition is satisfied by a power of the mapping and for which the power depends on the specified point in the space. We first state the result in the continuous case and later, show that the continuity is indeed not necessary. 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