Metric Spaces Research Articles

On (para)topological groups with a countably (s-)complete sieve

Countably sieve-complete spaces were defined by M. Michael in 1972. In this article, we introduce a notion called countably sieve-s-complete spaces. Some properties of countably sieve-complete (countably sieve-s-complete, strongly countably complete) spaces are discussed. We get the following main results.A topological group is countably sieve-complete if and only if G contains a closed countably compact subgroup H such that the quotient space G/H is completely metrizable and the canonical quotient mapping π:G→G/H is closed. By the above conclusion and a known conclusion, we get that a topological group G is strongly countably complete if and only if G is countably sieve-complete. A topological group G is countably sieve-s-complete if and only if G contains a sequentially compact closed subgroup H with a countable base of open neighborhoods such that the quotient space G/H is a completely metrizable space and the canonical quotient mapping π:G→G/H is closed. An ω-balanced topological group G is countably sieve-complete if and only if G contains a closed countably compact invariant subgroup H such that the quotient space G/H is a completely metrizable topological group and the canonical quotient mapping π:G→G/H is closed. A topological group G is ω-narrow and countably sieve-s-complete if and only if G contains a sequentially compact closed invariant subgroup H with a countable base of open neighborhoods such that the quotient space G/H is a completely metrizable second-countable topological group and the canonical quotient mapping π:G→G/H is closed.We show that if G is a regular countably sieve-complete semitopological group with Sm(G)≤ω and satisfies property (⁎), then G is a topological group. Every regular totally ω-narrow countably sieve-complete paratopological group is a topological group.

Is the Free Locally Convex Space L(X) Nuclear?

Given a class $${{\mathcal {P}}}$$ of Banach spaces, a locally convex space (LCS) E is called multi- $${{\mathcal {P}}}$$ if E can be isomorphically embedded into a product of spaces that belong to $${{\mathcal {P}}}$$ . We investigate the question whether the free locally convex space L(X) is strongly nuclear, nuclear, Schwartz, multi-Hilbert or multi-reflexive. If X is a Tychonoff space containing an infinite compact subset then, as it follows from the results of Außenhofer (Topol Appl 134:90–102, 2007), L(X) is not nuclear. We prove that for such X the free LCS L(X) has the stronger property of not being multi-Hilbert. We deduce that if X is a k-space, then the following properties are equivalent: (1) L(X) is strongly nuclear; (2) L(X) is nuclear; (3) L(X) is multi-Hilbert; (4) X is countable and discrete. On the other hand, we show that L(X) is strongly nuclear for every projectively countable P-space (in particular, for every Lindelöf P-space) X. We observe that every Schwartz LCS is multi-reflexive. It is known that if X is a $$k_\omega $$ -space, then L(X) is a Schwartz LCS (Außenhofer et al. in Stud Math 181(3):199–210, 2007), hence L(X) is multi-reflexive. We show that for every first-countable paracompact (in particular, for every metrizable) space X the converse is true, so L(X) is multi-reflexive if and only if X is a $$k_\omega $$ -space, equivalently if X is a locally compact and $$\sigma $$ -compact space. Similarly, we show that for any first-countable paracompact space X the free abelian topological group A(X) is a Schwartz group if and only if X is a locally compact space such that the set $$X^{(1)}$$ of all non-isolated points of X is $$\sigma $$ -compact.

Covering versus partitioning with the Cantor space

What topological spaces can be partitioned into copies of the Cantor space 2ω? An obvious necessary condition is that a space can be partitioned into copies of 2ω only if it can be covered with copies of 2ω. We prove three theorems concerning when this necessary condition is also sufficient.If X is a metrizable space and |X|≤c+ω (the least limit cardinal above c), then X can be partitioned into copies of 2ω if and only if X can be covered with copies of 2ω. To show this cardinality bound is sharp, we construct a metrizable space of size c+(ω+1) that can be covered with copies of 2ω, but not partitioned into copies of 2ω.Similarly, if X is first countable and |X|≤c, then X can be partitioned into copies of 2ω if and only if X can be covered with copies of 2ω. On the other hand, there is a first countable space of size c+ that can be covered with copies of 2ω, but not partitioned into copies of 2ω.Finally, we show that, regardless of its cardinality, a completely metrizable space can be partitioned into copies of 2ω if and only if it can be covered with copies of 2ω if and only if it has no isolated points.

Some Results of Unique Common Fixed Point Generalized In Extended b_2-metric spaces

Recently, extended -metric space has been used as a generalization of both -metric and extended metric spaces. The object of this study is to evidence some findings of fixed point and common fixed point for mappings in the frame of extended -metric space, in addition, to generalizing some examples. The outcomes of this study are to generalize and extend some existing findings in previous literature.

CLASSIFICATION OF ONE DIMENSIONAL DYNAMICAL SYSTEMS BY COUNTABLE STRUCTURES

AbstractWe study the complexity of the classification problem of conjugacy on dynamical systems on some compact metrizable spaces. Especially we prove that the conjugacy equivalence relation of interval dynamical systems is Borel bireducible to isomorphism equivalence relation of countable graphs. This solves a special case of Hjorth’s conjecture which states that every orbit equivalence relation induced by a continuous action of the group of all homeomorphisms of the closed unit interval is classifiable by countable structures. We also prove that conjugacy equivalence relation of Hilbert cube homeomorphisms is Borel bireducible to the universal orbit equivalence relation.

Baire property of some function spaces

A compact space X is called \(\pi\)-monolithic if for any surjective continuous mapping \(f \colon X \rightarrow K\) where K is a metrizable compact space there exists a metrizable compact space \(T \subseteq X\) such that \(f(T)=K\). A topological space X is Baire if the intersection of any sequence of open dense subsets of X is dense in X. Let \(C_p(X, Y)\) denote the space of all continuous Y-valued functions C(X,Y) on a Tychonoff space X with the topology of pointwise convergence. In this paper we have proved that for a totally disconnected space X the space \(C_p(X,\{0,1\})\) is Baire if, and only if, \(C_p(X,K)\) is Baire for every \(\pi\)-monolithic compact space K.For a Tychonoff space X the space \(C_p(X,\mathbb{R})\) is Baire if, and only if, \(C_p(X,L)\) is Baire for each Fréchet space L.We construct a totally disconnected Tychonoff space T such that \(C_p(T,M)\) is Baire for a separable metric space M if, and only if, M is a Peano continuum. Moreover, \(C_p(T,[0,1])\) is Baire but \(C_p(T,\{0,1\})\) is not.

Common Fixed Point Theorems for Four Weakly Compatible Self Maps Along with (CLR) Property in Fuzzy 2-Metric Spaces

In this paper, we prove some common fixed point theorems for four weakly compatible self-maps along with (CLR) property in fuzzy 2- metric spaces. Our results are the improved version of the theorems proved by Shojaei et al. [1] in 2013, since our results does not require closedness of ranges of subsets of X.

Random walks on convergence groups

We extend some properties of random walks on hyperbolic groups to random walks on convergence groups. In particular, we prove that if a convergence group G acts on a compact metrizable space M with the convergence property, then we can provide G\cup M with a compact topology such that random walks on G converge almost surely to points in M . Furthermore, we prove that if G is finitely generated and the random walk has finite entropy and finite logarithmic moment with respect to the word metric, then M , with the corresponding hitting measure, can be seen as a model for the Poisson boundary of G .

Weak Mean Equicontinuity for a Countable Discrete Amenable Group Action

The weak mean equicontinuity for a countable discrete amenable group G acting continuously on a compact metrizable space X is studied. It is shown that weak mean equicontinuity, continuously pointwise ergodicity and uniformity are coincided. Moreover, we prove that (X, G) is mean equicontinuous if and only if the product system \((X \times X,G)\) is weak mean equicontinuous.

On continuity of set-valued mappings

We will show that under certain circumstances upper and lower quasi-continuity implies continuity. We also extend Fort's theorem for upper quasi-continuous mappings, by showing that every upper quasi-continuous mapping from a Baire space into compact subsets of a regular locally metrizable space is continuous at points of a dense Gδ subset of X.

Directional mean dimension and continuum-wise expansive ℤ^{𝕜}-actions

We study directional mean dimension of Z k \mathbb {Z}^k -actions (where k k is a positive integer). On the one hand, we show that there is a Z 2 \mathbb {Z}^2 -action whose directional mean dimension (considered as a [ 0 , + ∞ ] [0,+\infty ] -valued function on the torus) is not continuous. On the other hand, we prove that if a Z k \mathbb {Z}^k -action is continuum-wise expansive, then the values of its ( k − 1 ) (k-1) -dimensional directional mean dimension are bounded. This is a generalization (with a view towards Meyerovitch and Tsukamoto’s theorem on mean dimension and expansive multiparameter actions) of a classical result due to Mañé: Any compact metrizable space admitting an expansive homeomorphism (with respect to a compatible metric) is finite-dimensional.

K-Spaces, sequential spaces and related topics in the absence of the axiom of choice

In the absence of the axiom of choice, among other new results concerning sequential, Fréchet-Urysohn, Loeb, Cantor completely metrizable, k-spaces and very k-spaces, it is proved that every Loeb T3-space having a base expressible as a countable union of finite sets is a metrizable second-countable space all of whose Fσ-subspaces are separable; every Gδ-subspace of a second-countable, Cantor completely metrizable space is Cantor completely metrizable, Loeb and separable. Arkhangel'skii's statement that every very k -space is Fréchet-Urysohn is unprovable in ZF but it holds in ZF that every first-countable, regular very k-space whose family of all non-empty compact sets has a choice function is Fréchet-Urysohn. That every second-countable metrizable space is a very k-space is equivalent to the axiom of countable choice for R.

Radius of comparison and mean cohomological independence dimension

We introduce a notion of mean cohomological independence dimension for actions of discrete amenable groups on compact metrizable spaces, as a variant of mean dimension, and use it to obtain lower bounds for the radius of comparison of the associated crossed product C⁎-algebras. Our general theory, gives the following for the minimal subshifts constructed by Dou in 2017. For any countable amenable group G and any polyhedron Z, Dou's subshift T of ZG with density parameter ρ satisfiesrc(C(X)⋊TG)>12mdim(T)(1−1−ρρ)−2.If k=dim⁡(Z) is even and Hˇk(Z;Q)≠0, thenrc(C(X)⋊TG)>12mdim(T)−1,regardless of what ρ is.

Chaotic behaviour of countable products of homeomorphism groups

ABSTRACT Relationships between a chaotic behaviour and closely related properties of topological transitivity, sensitivity to initial conditions, density of closed orbits of homeomorphism groups and their countable products are investigated. We provide a large number of new examples of chaotic groups of homeomorphisms of countable products of various metrizable topological spaces, including infinite-dimensional topological manifolds, whose factors can be as noncompact surfaces, so triangulable closed manifolds of an arbitrary dimension.

Minimal direct products

A space is called minimal if it admits a minimal continuous selfmap. We give examples of metrizable continua $X$ admitting both minimal homeomorphisms and minimal noninvertible maps, whose squares $X\times X$ are not minimal, i.e., they admit neither minimal homeomorphisms nor minimal noninvertible maps, thus providing a definitive answer to a question posed by Bruin, Kolyada and the second author in 2003. (In 2018, Boronski, Clark and Oprocha provided an answer in the case when only homeomorphisms were considered.) Then we introduce and study the notion of product-minimality. We call a compact metrizable space $Y$ product-minimal if, for every minimal system $(X,T)$ given by a metrizable space $X$ and a continuous selfmap $T$, there is a continuous map $S\colon Y\to Y$ such that the product $(X\times Y,T\times S)$ is minimal. If such a map $S$ always exists in the class of homeomorphisms, we say that $Y$ is a homeo-product-minimal space. We show that many classical examples of minimal spaces, including compact connected metrizable abelian groups, compact connected manifolds without boundary admitting a free action of a nontrivial compact connected Lie group, and many others, are in fact homeo-product-minimal.

Polynomial growth, comparison, and the small boundary property

We show that a minimal action of a finitely generated group of polynomial growth on a compact metrizable space has comparison. It follows that if such an action is free and has the small boundary property then it is almost finite and its C⁎-crossed product is Z-stable, and consequently that such crossed products are classified by their Elliott invariant.

Approximating the group algebra of the lamplighter by infinite matrix products

AbstractIn this paper, we introduce a new technique in the study of the*{*}-regular closure of some specific group algebrasKGinside𝒰⁢(G){{\mathcal{U}}(G)}, the*{*}-algebra of unbounded operators affiliated to the group von Neumann algebra𝒩⁢(G){{\mathcal{N}}(G)}. The main tool we use for this study is a general approximation result for a class of crossed product algebras of the formCK⁢(X)⋊Tℤ{C_{K}(X)\rtimes_{T}{\mathbb{Z}}}, whereXis a totally disconnected compact metrizable space,Tis a homeomorphism ofX, andCK⁢(X){C_{K}(X)}stands for the algebra of locally constant functions onXwith values on an arbitrary fieldK. The connection between this class of algebras and a suitable class of group algebras is provided by the Fourier transform. Utilizing this machinery, we study an explicit approximation for the lamplighter group algebra. This is used in another paper by the authors to obtain a whole family ofℓ2{\ell^{2}}-Betti numbers arising from the lamplighter group, most of them transcendental.

Entropy pair realization

AbstractWe show that the complete positive entropy (CPE) class $\alpha $ of Barbieri and García-Ramos contains a one-dimensional subshift for all countable ordinals $\alpha $ , that is, the process of alternating topological and transitive closure on the entropy pairs relation of a subshift can end on an arbitrary ordinal. This is the composition of three constructions. We first realize every ordinal as the length of an abstract ‘close-up’ process on a countable compact space. Next, we realize any abstract process on a compact zero-dimensional metrizable space as the process started from a shift-invariant relation on a subshift, the crucial construction being the implementation of every compact metrizable zero-dimensional space as an open invariant quotient of a subshift. Finally, we realize any shift-invariant relation E on a subshift X as the entropy pair relation of a supershift $Y \supset X$ , and under strong technical assumptions, we can make the CPE process on Y end on the same ordinal as the close-up process of E.

Garden of Eden and weakly periodic points for certain expansive actions of groups

AbstractWe present several applications of the weak specification property and certain topological Markov properties, recently introduced by Barbieri, García-Ramos, and Li [Markovian properties of continuous group actions: algebraic actions, entropy and the homoclinic group. Adv. Math.397 (2022), 52], and implied by the pseudo-orbit tracing property, for general expansive group actions on compact spaces. First we show that any expansive action of a countable amenable group on a compact metrizable space satisfying the weak specification and strong topological Markov properties satisfies the Moore property, that is, every surjective endomorphism of such dynamical system is pre-injective. This together with an earlier result of Li (where the strong topological Markov property is not needed) of the Myhill property [Garden of Eden and specification. Ergod. Th. & Dynam. Sys.39 (2019), 3075–3088], which we also re-prove here, establishes the Garden of Eden theorem for all expansive actions of countable amenable groups on compact metrizable spaces satisfying the weak specification and strong topological Markov properties. We hint how to easily generalize this result even for uncountable amenable groups and general compact, not necessarily metrizable, spaces. Second, we generalize the recent result of Cohen [The large scale geometry of strongly aperiodic subshifts of finite type. Adv. Math.308 (2017), 599–626] that any subshift of finite type of a finitely generated group having at least two ends has weakly periodic points. We show that every expansive action of such a group having a certain Markov topological property, again implied by the pseudo-orbit tracing property, has a weakly periodic point. If it has additionally the weak specification property, the set of such points is dense.

Some Results of Conditionally Sequential Absorbing Maps in 2-Menger Space

This paper aims to present two results in probabilistic 2-metric space by using the concepts of conditionally sequential absorbing maps and reciprocally continuous mappings. These are generalizations of the theorem proved by Gupta et al. [5]. Further, these are justified by supportingexamples.