Metric Spaces Research Articles

Some Results on Conditionally Compatible and Conditionally Semi-Compatible Mappings in Probabilistic 2-Metric Space

The objective of this paper is to obtain two results in probabilistic 2-metric space by employing the concepts of (E.A)-property, conditionally compatible, conditionally semi-compatible and sub-sequentially continuous mappings. These findings improve the theorem proved in [5]. Further, these results are substantiated by supporting examples.

Contractions with Comparison Function in the Extended Cone 2-Metric Space

The present paper deals with results on FP in a C-2MS using contractions and comparison function in setting of this. Outcomes improve and generalize a number of FP results in the literature. Fixed point theory is an important branch of mathematics related with functional analysis and topology. Fixed point theory is any several of theorems in mathematics dealing with a transformation of the point of a set into itself such that at least one point remains fixed as square of any real number no fixed except 0 and 1. i.e., 0 and 1 are only fixed points whose square is fixed in real number system. Among hundreds of fixed point theorems, Brouwer's is particularly well-known, due in part to its use across numerous fields of mathematics. In its original field, this result is one of the key theorems characterizing the topology of Euclidean spaces. The theorem is also used for proving deep results about differential equations and is covered in most introductory courses on differential geometry.

Some Results in Fuzzy 2- Metric Space

The notion of fuzzy 2-metric space and its essential features are initially examined in this study. Following that, the ideas of compact fuzzy2- metric space and fuzzy totally bounded space are discussed. Furthermore, the essential theorems relating to these concepts are established. In particular, we prove that a fuzzy 2- metric space which is compact must be complete.

A𝕋-algebras from fiberwise essentially minimal zero-dimensional dynamical systems

We introduce a type of zero-dimensional dynamical system (a pair consisting of a totally disconnected compact metrizable space along with a homeomorphism of that space), which we call “fiberwise essentially minimal”, that is a class that includes essentially minimal systems and systems in which every orbit is minimal. We prove that the associated crossed product [Formula: see text]-algebra of such a system is an A[Formula: see text]-algebra. Under the additional assumption that the system has no periodic points, we prove that the associated crossed product [Formula: see text]-algebra has real rank zero, which tells us that such [Formula: see text]-algebras are classifiable by [Formula: see text]-theory. The associated crossed product [Formula: see text]-algebras to these nontrivial examples are of particular interest because they are non-simple (unlike in the minimal case).

Fixed Point Theorems in Partial 2-Metric Spaces

In this paper we aim to introduce the concept of partial 2-metric spaces and study some fixed point theorems for self mappings defined on partial 2-metric spaces.

On a topological Ramsey theorem

AbstractWe introduce natural strengthenings of sequential compactness, the r-Ramsey property for each natural number $r\geq 1$ . We prove that metrizable compact spaces are r-Ramsey for all r and give examples of compact spaces that are r-Ramsey but not $(r+1)$ -Ramsey for each $r\geq 1$ (assuming Continuum Hypothesis (CH) for all $r>1$ ). Productivity of the r-Ramsey property is considered.

Topological Transcendental Fields

This article initiates the study of topological transcendental fields F which are subfields of the topological field C of all complex numbers such that F only consists of rational numbers and a nonempty set of transcendental numbers. F, with the topology it inherits as a subspace of C, is a topological field. Each topological transcendental field is a separable metrizable zero-dimensional space and algebraically is Q(T), the extension of the field of rational numbers by a set T of transcendental numbers. It is proven that there exist precisely 2ℵ0 countably infinite topological transcendental fields and each is homeomorphic to the space Q of rational numbers with its usual topology. It is also shown that there is a class of 22ℵ0 of topological transcendental fields of the form Q(T) with T a set of Liouville numbers, no two of which are homeomorphic.

Continua in the Gromov–Hausdorff space

We first prove that for all compact metrizable spaces, there exists a topological embedding of the compact metrizable space into each of the sets of compact metric spaces which are connected, path-connected, geodesic, or CAT(0), in the Gromov–Hausdorff space with finite prescribed values. As its application, we show that the sets prescribed above are path-connected and their non-empty open subsets have infinite topological dimension. By the same method, we also prove that the set of all proper CAT(0) spaces is path-connected and its non-empty open subsets have infinite topological dimension with respect to the pointed Gromov–Hausdorff distance.

Some Results of Conditionally Sequential Absorbing and Pseudo Reciprocally Continuous Mappings in Probabilistic 2-Metric Space

The objective of this paper is to generate two results in probabilistic 2-metric space by using the concepts of conditionally sequential absorbing mappings and pseudo reciprocally continuous mappings. These results stand as generalizations of the theorem proved by V. K. Gupta, Arihant Jain and Rajesh Kumar. Further these two outcomes are justified by supporting examples.

Generalization Fixed Points of Multivalued 𝜶-Admissible Mappings in 2-Metric Spaces

In this paper, we create some fixed-point results of multivalue α-admissible of 2-metric spaces. We introduce Hausdorff distance in 2-metric space, use it in our theorem. we investigated the existence of some fixed point results for new types of contraction. We study the stability of fixed point set.

Spaces where all closed sets are α-limit sets

Metrizable spaces are studied in which every closed set is an α-limit set for some continuous map and some point. It is shown that this property is enjoyed by every space containing sufficiently many arcs (formalized in the notion of a space with enough arcs), though such a space need not be arcwise connected. Further it is shown that this property is not preserved by topological sums, products and continuous images and quotients. However, positive results do hold for metrizable spaces obtained by those constructions from spaces with enough arcs.

On the widest class of completeness-preserving covering maps

We begin with the study of H-covering maps. H-covering maps between separable metrizable spaces are determined by their behaviour on some countable compact sets, and are invariant under compositions. H-covering continuous maps preserve Polish spaces.

Generalized a-min special type contraction results on 2-Menger spaces

The goal of this paper is to present some novel probabilistic a-minimum contraction results on probabilistic 2-metric spaces. Our findings are based on probabilistic 2-metric spaces, which are the probabilistic generalisations of 2-metric spaces. An illustrative example backs up our findings.

Interpolative contractions and intuitionistic fuzzy set-valued maps with applications

<abstract><p>Over time, the interpolative approach in fixed point theory (FPT) has been investigated only in the setting of crisp mathematics, thereby dropping-off a significant amount of useful results. As an attempt to fill up the aforementioned gaps, this paper initiates certain hybrid concepts under the names of interpolative Hardy-Rogers-type (IHRT) and interpolative Reich-Rus-Ciric type (IRRCT) intuitionistic fuzzy contractions in the frame of metric space (MS). Adequate criteria for the existence of intuitionistic fuzzy fixed point (FP) for such contractions are examined. On the basis that FP of a single-valued mapping obeying interpolative type contractive inequality is not always unique, and thereby making the ideas more suitable for FP theorems of multi-valued mappings, a few special cases regarding point-to-point and non-fuzzy set-valued mappings which include the conclusions of some well-known results in the corresponding literature are highlighted and discussed. In addition, comparative examples which dwell on the generality of our obtained results are constructed.</p></abstract>

Application of Common Fixed Point Theorem on Revised Fuzzy Metric Space

In this article, we introduces the concept of the common fixed point theorem for three function in the revised fuzzy metric space for different applications in the revised fuzzy 2-metric space and revised fuzzy 3-metric with examples. Here, too, the result is generalized and improved.

Some independence results about compact metrizable spaces and two notions of finiteness

We settle some open problems from Keremedis, Tachtsis and Wajch (2021) [16] and from Keremedis and Tachtsis (2001) [15].In particular, we establish the following results:1.There is a permutation model of ZFA in which every linearly ordered set can be well ordered, the axiom of choice for denumerable families of non-empty finite sets and the axiom of countable multiple choice are false, and there exists a compact, dense-in-itself, non-second countable, weakly separable, metrizable space (X,τ) with a base for τ that can be written as a union of a denumerable family of finite sets, but the family of all non-empty closed subsets of X has no multiple choice function in the model.2.There is a model of ZF which has the same properties as the model of (1), except for “every linearly ordered set can be well ordered” which is false in the model.3.There is a model of ZF in which there is a filter base infinite set X which is strongly filter base finite, and thus ω+1 with the order topology is not a remainder of the discrete space (X,P(X)) in the model. For the independence results (2) and (3), we will use certain permutation models of ZFA and then we will apply the Jech–Sochor First Embedding Theorem in order to transfer those results into ZF. Specifically, the model of (1) (recently constructed by Howard and Tachtsis ((2021) [8])) will be used for (2), while for (3) we will introduce a new permutation model and prove that it satisfies the proposition given in (3).

A Result on probabilistic 2-metric space using strong semi compatible mappings

Objective/Aim: To generate a fixed point theorem in probabilistic 2-metric space. Method: By employing strong semi compatible mappings and sub sequentially continuous mappings. Findings: Generated unique common fixed point theorem and substantiated with appropriate example. Novelty/ Improvement: The concepts of strong semi compatible mappings and sub sequentially continuous mappings are weaker than existing conditions like weakly compatible mappings which generalizes the theorem of V. K. Gupta, Arihant Jain and Rajesh kumar. Keywords: Strong semi compatible; sub sequentially continuous; Probabilistic 2metric space; conditional semi compatible; conditional compatible

On some unique fixed point theorems with rational expressions in partially ordered metric spaces

The aim of this paper is to establish some fixed point results of a self mapping satisfying generalized rational contraction conditions in a metric space endowed with a partial order. These results generalized and widen the result of Sharma and Yuel from metrical space to partially ordered metric space and also a few well known results in an ordered metric space. Some examples are illustrated to support the usability of the presented results.

Entropy, products, and bounded orbit equivalence

AbstractWe prove that if two topologically free and entropy regular actions of countable sofic groups on compact metrizable spaces are continuously orbit equivalent, and each group either (i) contains a w-normal amenable subgroup which is neither locally finite nor virtually cyclic, or (ii) is a non-locally-finite product of two infinite groups, then the actions have the same sofic topological entropy. This fact is then used to show that if two free uniquely ergodic and entropy regular probability-measure-preserving actions of such groups are boundedly orbit equivalent then the actions have the same sofic measure entropy. Our arguments are based on a relativization of property SC to sofic approximations and yield more general entropy inequalities.

COMMUNAL FIXED POINT IN FUZZY 2- METRIC SPACES

The objective of this paper is to emphasize the Common Fixed Point in Fuzzy 2- Metric Spaces and prove a common fixed point theorem of compatible mappings of type (R) in fuzzy 2-metric space. We consider four mappings of which one is continuous. The results generalize many results in the literature. Some illustrative examples are furnished which demonstrate the validity of the hypotheses and the degree of utility of our results.