Metric Spaces Research Articles

Almost finiteness, comparison, and tracial [formula omitted]-stability

Inspired by Kerr's work on topological dynamics, we define tracial Z-stability for sub-C⁎-algebras. We prove that for a countable discrete amenable group G acting freely and minimally on a compact metrizable space X, tracial Z-stability for the sub-C⁎-algebra (C(X)⊆C(X)⋊G) implies that the action has dynamical comparison. Consequently, tracial Z-stability is equivalent to almost finiteness of the action, provided that the action has the small boundary property.

Some Results by Using CLR’s-Property in Probabilistic 2-Metric Space

The aim of this paper is to generate two fixed point theorems in probabilistic 2-metric space by applying CLR’S-property and occasionally weakly compatible mappings (OWC), these two results generalize the theorem proved by V. K. Gupta, Arihant Jain and Rajesh Kumar. Further these results are justified with suitable examples.

Non-normality points and nice spaces

J. Terasawa in "$\beta X-\{p\}$ are non-normal for non-discrete spaces $X$" (2007) and the author in "On non-normality points and metrizable crowded spaces" (2007), independently showed for any metrizable crowded space $X$ that each point $p$ of its Čech--Stone remainder $X^{*}$ is a non-normality point of $\beta X$. We introduce a new class of spaces, named nice spaces, which contains both of Sorgenfrey line and every metrizable crowded space. We obtain the result above for every nice space.

Multiplier Algebras of Normed Spaces of Continuous Functions

In this article, we investigate some general properties of the multiplier algebras of normed spaces of continuous functions (NSCF). In particular, we prove that the multiplier algebra inherits some of the properties of the NSCF. We show that it is often possible to construct NSCF’s which only admit constant multipliers. To do that, using a method from Mashreghi and Ransford (Anal Math Phys 9(2):899–905, 2019), we prove that any separable Banach space can be realized as a NSCF over any separable metrizable space of infinite cardinality. On the other hand, we give a sufficient condition for non-separability of a multiplier algebra.

Chaos on Fuzzy Dynamical Systems

Given a continuous map f:X→X on a metric space, it induces the maps f¯:K(X)→K(X), on the hyperspace of nonempty compact subspaces of X, and f^:F(X)→F(X), on the space of normal fuzzy sets, consisting of the upper semicontinuous functions u:X→[0,1] with compact support. Each of these spaces can be endowed with a respective metric. In this work, we studied the relationships among the dynamical systems (X,f), (K(X),f¯), and (F(X),f^). In particular, we considered several dynamical properties related to chaos: Devaney chaos, A-transitivity, Li–Yorke chaos, and distributional chaos, extending some results in work by Jardón, Sánchez and Sanchis (Mathematics 2020, 8, 1862) and work by Bernardes, Peris and Rodenas (Integr. Equ. Oper. Theory 2017, 88, 451–463). Especial attention is given to the dynamics of (continuous and linear) operators on metrizable topological vector spaces (linear dynamics).

Ergodic Decomposition of Dirichlet Forms via Direct Integrals and Applications

We study direct integrals of quadratic and Dirichlet forms. We show that each quasi-regular Dirichlet space over a probability space admits a unique representation as a direct integral of irreducible Dirichlet spaces, quasi-regular for the same underlying topology. The same holds for each quasi-regular strongly local Dirichlet space over a metrizable Luzin σ-finite Radon measure space, and admitting carré du champ operator. In this case, the representation is only projectively unique.

Basic properties of 𝑋 for which the space 𝐶_{𝑝}(𝑋) is distinguished

In our paper [Proc. Amer. Math. Soc. Ser. B 8 (2021), pp. 86–99] we showed that a Tychonoff spaceXXis aΔ\Delta-space (in the sense of R. W. Knight [Trans. Amer. Math. Soc. 339 (1993), pp. 45–60], G. M. Reed [Fund. Math. 110 (1980), pp. 145–152]) if and only if the locally convex spaceCp(X)C_{p}(X)is distinguished. Continuing this research, we investigate whether the classΔ\DeltaofΔ\Delta-spaces is invariant under the basic topological operations.We prove that ifX∈ΔX \in \Deltaandφ:X→Y\varphi :X \to Yis a continuous surjection such thatφ(F)\varphi (F)is anFσF_{\sigma }-set inYYfor every closed setF⊂XF \subset X, then alsoY∈ΔY\in \Delta. As a consequence, ifXXis a countable union of closed subspacesXiX_isuch that eachXi∈ΔX_i\in \Delta, then alsoX∈ΔX\in \Delta. In particular,σ\sigma-product of any family of scattered Eberlein compact spaces is aΔ\Delta-space and the product of aΔ\Delta-space with a countable space is aΔ\Delta-space. Our results give answers to several open problems posed by us [Proc. Amer. Math. Soc. Ser. B 8 (2021), pp. 86–99].LetT:Cp(X)⟶Cp(Y)T:C_p(X) \longrightarrow C_p(Y)be a continuous linear surjection. We observe thatTTadmits an extension to a linear continuous operatorT^\widehat {T}fromRX\mathbb {R}^XontoRY\mathbb {R}^Yand deduce thatYYis aΔ\Delta-space wheneverXXis. Similarly, assuming thatXXandYYare metrizable spaces, we show thatYYis aQQ-set wheneverXXis.Making use of obtained results, we provide a very short proof for the claim that every compactΔ\Delta-space has countable tightness. As a consequence, under Proper Forcing Axiom every compactΔ\Delta-space is sequential.In the article we pose a dozen open questions.

New Results on the Aggregation of Norms

It is a natural question if a Cartesian product of objects produces an object of the same type. For example, it is well known that a countable Cartesian product of metrizable topological spaces is metrizable. Related to this question, Borsík and Doboš characterized those functions that allow obtaining a metric in the Cartesian product of metric spaces by means of the aggregation of the metrics of each factor space. This question was also studied for norms by Herburt and Moszyńska. This aggregation procedure can be modified in order to construct a metric or a norm on a certain set by means of a family of metrics or norms, respectively. In this paper, we characterize the functions that allow merging an arbitrary collection of (asymmetric) norms defined over a vector space into a single norm (aggregation on sets). We see that these functions are different from those that allow the construction of a norm in a Cartesian product (aggregation on products). Moreover, we study a related topological problem that was considered in the context of metric spaces by Borsík and Doboš. Concretely, we analyze under which conditions the aggregated norm is compatible with the product topology or the supremum topology in each case.

On the continuity of the elements of the Ellis semigroup and other properties

We consider discrete dynamical systems whose phase spaces are compact metrizable countable spaces. In the first part of the article, we study some properties that guarantee the continuity of all functions of the corresponding Ellis semigroup. For instance, if every accumulation point of $X$ is fixed, we give a necessary and sufficient condition on a point $a\in X'$ in order that all functions of the Ellis semigroup $E(X,f)$ be continuous at the given point $a$. In the second part, we consider transitive dynamical systems. We show that if $(X,f)$ is a transitive dynamical system and either every function of $E(X,f)$ is continuous or $|\omega_f(x)|=1$ for each accumulation point $x$ of $X$, then $E(X,f)$ is homeomorphic to $X$. Several examples are given to illustrate our results.

On some inequalities in 2-metric spaces

In this paper, we establish new inequalities in the setting of 2-metric spaces and provide their geometric interpretations. Some of our results are extensions of those obtained by Dragomir and Goşa (J. Indones. Math. Soc. 11(1):33–38, 2005) in the setting of metric spaces.

METRIC SPACE AND ITS APPLICATIONS

This paper gives a short review of some basics of metric spaces, the concept of continuity of these metric spaces, with a comparison of it with uniform continuity and convergence. In addition to this, it includes a discussion of some past work of researchers on metric space, its generalization, and useful results made by them with applications using these spaces.

Dislocated quasi rectangular $b$-metric spaces and fixed point theorems of cyclic and weakly cyclic contractions

In this paper we have established fixed point theorems for $dq$-rectangular $b$-cyclic-Banach and $dq$-rectangular $b$-cyclic-Kannan contraction mappings in dislocated quasi rectangular $b$-metric spaces. We have also presented examples to support some of our results.

Several results on compact metrizable spaces in mathbf {ZF}

In the absence of the axiom of choice, the set-theoretic status of many natural statements about metrizable compact spaces is investigated. Some of the statements are provable in mathbf {ZF}, some are shown to be independent of mathbf {ZF}. For independence results, distinct models of mathbf {ZF} and permutation models of mathbf {ZFA} with transfer theorems of Pincus are applied. New symmetric models of mathbf {ZF} are constructed in each of which the power set of mathbb {R} is well-orderable, the Continuum Hypothesis is satisfied but a denumerable family of non-empty finite sets can fail to have a choice function, and a compact metrizable space need not be embeddable into the Tychonoff cube [0, 1]^{mathbb {R}}.

Conditional variational principles of conditional entropies for amenable group actions * *The work was supported by NSF of China (11671058, 12071049) and 2020CDCGJ019.

Let G be an infinite discrete countable amenable group acting continuously on two compact metrizable spaces X, Y. Assume that φ : (Y, G) → (X, G) is a factor map. Using finite open covers, the conditional topological entropy of φ is defined. The conditional measure-theoretic entropy of φ equals the conditional measure-theoretic entropy of Y to X. With the aid of tiling system of G, the conditional variational principle of φ is studied when (X, G) is an asymptotically h-expansive system. If X = Y and φ is the identity map, the conditional topological entropy of system (X, G) is defined. In the Cartesian square (X × X, G), we define the conditional measure-theoretic entropy of (X, G) to be the defect of the upper semi-continuity of the conditional measure-theoretic entropy of X × X to the first axis. Then the conditional variational principle of (X, G) is obtained.

New coupled and common coupled fixed point results with generalized c -distance on cone b -metric spaces

In this paper, we prove the existence and uniqueness of common coupled fixed point and coupled fixed point in cone \({b}\)-metric spaces with generalized \({c}\)-distance. Our results extend and generalize several well-known comparable results in literature. We provide one example to support our obtained results.

There is no compact metrizable space containing all continua as unique components

We answer a question of Piotr Minc by proving that there is no compact metrizable space whose set of components contains a unique topological copy of every metrizable compactification of a ray (i.e. a half-open interval) with an arc (i.e. closed bounded interval) as the remainder. To this end we use the concept of Borel reductions coming from Invariant descriptive set theory. It follows as a corollary that there is no compact metrizable space such that every continuum is homeomorphic to exactly one component of this space.

Second-countable compact Hausdorff spaces as remainders in ZF and two new notions of infiniteness

In the absence of the Axiom of Choice, necessary and sufficient conditions for a locally compact Hausdorff space to have all non-empty second-countable compact Hausdorff spaces as remainders are given in ZF. Among other independence results, the characterization of locally compact Hausdorff spaces having all non-empty metrizable compact spaces as remainders, obtained by Hatzenbuhler and Mattson in ZFC, is proved to be independent of ZF. Urysohn's Metrization Theorem is generalized. New concepts of a strongly filterbase infinite set and a dyadically filterbase infinite set are introduced, both stemming from the investigations on compactifications. Set-theoretic and topological definitions of the new concepts are given, and their relationship with certain known notions of infinite sets is investigated in ZF. A new permutation model is introduced in which there exists a strongly filterbase infinite set which is weakly Dedekind-finite.

Some Common Fixed Point Results for Generalized $alpha_*$-$psi$-contractive Multi-valued Mappings on Ordered Metric Spaces with Application to Initial Value Problem

In 2012, Samet, et al. introduced the notion of $alpha$-$psi$-contractive type mappings. They have been establish some fixed point theorems for the mappings on complete metricspaces. In this paper, we introduce the notion of generalized $alpha_*$-$psi$-contractive multi-valued mappings and we give some related fixed point results on ordered metric spaces via application to an initial value problem.

Domination in the sense of diagrams

In this article the notion of domination of metrizable spaces in the sense of diagrams has been introduced. A few of its properties and applications are given. It deserves particular attention that the new domination can be applied to the coincidence theory.

Fences, their endpoints, and projective Fraïssé theory

We introduce a new class of compact metrizable spaces, which we call fences, and its subclass of smooth fences. We isolate two families F , F 0 \mathcal {F}, \mathcal {F}_{0} of Hasse diagrams of finite partial orders and show that smooth fences are exactly the spaces which are approximated by projective sequences from F 0 \mathcal {F}_{0} . We investigate the combinatorial properties of Hasse diagrams of finite partial orders and show that F , F 0 \mathcal {F}, \mathcal {F}_{0} are projective Fraïssé families with a common projective Fraïssé limit. We study this limit and characterize the smooth fence obtained as its quotient, which we call a Fraïssé fence. We show that the Fraïssé fence is a highly homogeneous space which shares several features with the Lelek fan, and we examine the structure of its spaces of endpoints. Along the way we establish some new facts in projective Fraïssé theory.