ON THE COMPLETION OF -METRIC SPACES

Based on the metrisation of $b$-metric spaces of Paluszyński and Stempak [‘On quasi-metric and metric spaces’, Proc. Amer. Math. Soc.137(12) (2009), 4307–4312], we prove that every $b$-metric space has a completion. Our approach resolves the limitation in using the quotient space of equivalence classes of Cauchy sequences to obtain a completion of a $b$-metric space.

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Cauchy Points of Metric Locales

Canadian Journal of Mathematics ◽

10.4153/cjm-1989-038-0 ◽

1989 ◽

Vol 41 (5) ◽

pp. 830-854 ◽

Cited By ~ 7

Author(s):

B. Banaschewski ◽

A. Pultr

Keyword(s):

Equivalence Relation ◽

Metric Spaces ◽

Equivalence Classes ◽

Classical Description ◽

Natural Approach ◽

Precise Meaning ◽

Natural Equivalence ◽

Cauchy Sequences

A natural approach to topology which emphasizes its geometric essence independent of the notion of points is given by the concept of frame (for instance [4], [8]). We consider this a good formalization of the intuitive perception of a space as given by the “places” of non-trivial extent with appropriate geometric relations between them. Viewed from this position, points are artefacts determined by collections of places which may in some sense by considered as collapsing or contracting; the precise meaning of the latter as well as possible notions of equivalence being largely arbitrary, one may indeed have different notions of point on the same “space”. Of course, the well-known notion of a point as a homomorphism into 2 evidently fits into this pattern by the familiar correspondence between these and the completely prime filters. For frames equipped with a diameter as considered in this paper, we introduce a natural alternative, the Cauchy points. These are the obvious counterparts, for metric locales, of equivalence classes of Cauchy sequences familiar from the classical description of completion of metric spaces: indeed they are decreasing sequences for which the diameters tend to zero, identified by a natural equivalence relation.

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On statistical convergence in quasi-metric spaces

Demonstratio Mathematica ◽

10.1515/dema-2019-0019 ◽

2019 ◽

Vol 52 (1) ◽

pp. 225-236 ◽

Cited By ~ 2

Author(s):

Merve İlkhan ◽

Emrah Evren Kara

Keyword(s):

Functional Analysis ◽

Computer Science ◽

Distance Function ◽

Metric Space ◽

Statistical Convergence ◽

Triangle Inequality ◽

Metric Spaces ◽

Applied Mathematics ◽

Comprehensive Investigation ◽

Cauchy Sequences

AbstractA quasi-metric is a distance function which satisfies the triangle inequality but is not symmetric in general. Quasi-metrics are a subject of comprehensive investigation both in pure and applied mathematics in areas such as in functional analysis, topology and computer science. The main purpose of this paper is to extend the convergence and Cauchy conditions in a quasi-metric space by using the notion of asymptotic density. Furthermore, some results obtained are related to completeness, compactness and precompactness in this setting using statistically Cauchy sequences.

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Coarse compactifications and controlled products

Journal of Topology and Analysis ◽

10.1142/s1793525321500102 ◽

2020 ◽

pp. 1-26

Author(s):

Tomohiro Fukaya ◽

Shin-ichi Oguni ◽

Takamitsu Yamauchi

Keyword(s):

Metric Space ◽

Metric Spaces ◽

Equivalence Classes ◽

Ideal Boundary ◽

Bijective Correspondence ◽

Gromov Boundary ◽

The Ideal

We introduce the notion of controlled products on metric spaces as a generalization of Gromov products, and construct boundaries by using controlled products, which we call the Gromov boundaries. It is shown that the Gromov boundary with respect to a controlled product on a proper metric space is the ideal boundary of a coarse compactification of the space. It is also shown that there is a bijective correspondence between the set of all coarse equivalence classes of controlled products and the set of all equivalence classes of coarse compactifications.

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On pseudo-valuations on BCK-algebras

Filomat ◽

10.2298/fil1812319m ◽

2018 ◽

Vol 32 (12) ◽

pp. 4319-4332 ◽

Cited By ~ 1

Author(s):

S. Mehrshad ◽

N. Kouhestani

Keyword(s):

Finite Number ◽

Metric Space ◽

Metric Spaces ◽

Quotient Space

In this paper, we study some properties of pseudo-valuations and their induced quasi metrics. The continuity of operation of a BCK-algebra was studied with topology induced by a pseudo-valuation. Moreover, we show that product of finite number of this pseudo metric spaces is a pseudo metric space. Also, we prove that if a BCK-algebra X has a pseudo-valuation, then every quotient space of X has a pseudo metric. The completion of this spaces has been investigated in the present study.

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Using the Supremum Form of Auxiliary Functions to Study the Common Coupled Coincidence Points in Fuzzy Semi-Metric Spaces

Axioms ◽

10.3390/axioms10010005 ◽

2021 ◽

Vol 10 (1) ◽

pp. 5

Author(s):

Hsien-Chung Wu

Keyword(s):

Metric Space ◽

Metric Spaces ◽

Coincidence Points ◽

Coupled Coincidence Points ◽

Cauchy Sequences ◽

The Common ◽

Rational Condition ◽

Common Coupled Fixed Points ◽

Coupled Coincidence ◽

Coupled Fixed Points

This paper investigates the common coupled coincidence points and common coupled fixed points in fuzzy semi-metric spaces. The symmetric condition is not necessarily satisfied in fuzzy semi-metric space. Therefore, four kinds of triangle inequalities are taken into account in order to study the Cauchy sequences. Inspired by the intuitive observations, the concepts of rational condition and distance condition are proposed for the purpose of simplifying the discussions.

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Strongly $\lambda$-Statistically and Strongly Vallée-Poussin Pre-Cauchy Sequences in Probabilistic Metric Spaces

Tamkang Journal of Mathematics ◽

10.5556/j.tkjm.53.2022.3893 ◽

2021 ◽

Vol 53 ◽

Author(s):

Argha Ghosh ◽

Samiran Das

Keyword(s):

Metric Space ◽

Metric Spaces ◽

Strong Topology ◽

Probabilistic Metric Space ◽

Probabilistic Metric Spaces ◽

Cauchy Sequences ◽

Probabilistic Metric ◽

Convergent Sequences

We introduce the notions of strongly $\lambda$-statistically pre-Cauchy and strongly Vall´ee-Poussin pre-Cauchy sequences in probabilistic metric spaces endowed with strong topology. And we show that these two new notions are equivalent. Strongly $\lambda$-statistically convergent sequences are strongly $\lambda$-statistically pre-Cauchy sequences, and we give an example to show that there is a sequence in a probabilistic metric space which is strongly $\lambda$-statistically pre-Cauchy but not strongly $\lambda$-statistically convergent.

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On Asymmetric Distances

Analysis and Geometry in Metric Spaces ◽

10.2478/agms-2013-0004 ◽

2013 ◽

Vol 1 ◽

pp. 200-231 ◽

Cited By ~ 6

Author(s):

Andrea C.G. Mennucci

Keyword(s):

Total Variation ◽

Calculus Of Variations ◽

Distance Function ◽

Metric Space ◽

Metric Spaces ◽

Geodesic Segment ◽

Intrinsic Metric ◽

Typical Application ◽

Variation Formula

Abstract In this paper we discuss asymmetric length structures and asymmetric metric spaces. A length structure induces a (semi)distance function; by using the total variation formula, a (semi)distance function induces a length. In the first part we identify a topology in the set of paths that best describes when the above operations are idempotent. As a typical application, we consider the length of paths defined by a Finslerian functional in Calculus of Variations. In the second part we generalize the setting of General metric spaces of Busemann, and discuss the newly found aspects of the theory: we identify three interesting classes of paths, and compare them; we note that a geodesic segment (as defined by Busemann) is not necessarily continuous in our setting; hence we present three different notions of intrinsic metric space.

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A new Ф-generalized quasi metric space with some fixed point results and applications

Filomat ◽

10.2298/fil1711157a ◽

2017 ◽

Vol 31 (11) ◽

pp. 3157-3172

Author(s):

Mujahid Abbas ◽

Bahru Leyew ◽

Safeer Khan

Keyword(s):

Fixed Point ◽

Differential Equations ◽

Fixed Points ◽

Periodic Solutions ◽

Metric Space ◽

Delay Differential Equations ◽

Fixed Point Theorems ◽

Metric Spaces ◽

Existence Of Periodic Solutions ◽

Delay Differential

In this paper, the concept of a new ?-generalized quasi metric space is introduced. A number of well-known quasi metric spaces are retrieved from ?-generalized quasi metric space. Some general fixed point theorems in a ?-generalized quasi metric spaces are proved, which generalize, modify and unify some existing fixed point theorems in the literature. We also give applications of our results to obtain fixed points for contraction mappings in the domain of words and to prove the existence of periodic solutions of delay differential equations.

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On Some New Contractive Conditions in Complete Metric Spaces

Mathematics ◽

10.3390/math9020118 ◽

2021 ◽

Vol 9 (2) ◽

pp. 118

Author(s):

Jelena Vujaković ◽

Eugen Ljajko ◽

Mirjana Pavlović ◽

Stojan Radenović

Keyword(s):

Current Literature ◽

Metric Space ◽

Metric Spaces ◽

Complete Metric Spaces ◽

Nonlinear Anal ◽

Increasing Mapping ◽

Strictly Increasing Mapping

One of the main goals of this paper is to obtain new contractive conditions using the method of a strictly increasing mapping F:(0,+∞)→(−∞,+∞). According to the recently obtained results, this was possible (Wardowski’s method) only if two more properties (F2) and (F3) were used instead of the aforementioned strictly increasing (F1). Using only the fact that the function F is strictly increasing, we came to new families of contractive conditions that have not been found in the existing literature so far. Assuming that α(u,v)=1 for every u and v from metric space Ξ, we obtain some contractive conditions that can be found in the research of Rhoades (Trans. Amer. Math. Soc. 1977, 222) and Collaco and Silva (Nonlinear Anal. TMA 1997). Results of the paper significantly improve, complement, unify, generalize and enrich several results known in the current literature. In addition, we give examples with results in line with the ones we obtained.

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An Intrinsic Characterization of Five Points in a CAT(0) Space

Analysis and Geometry in Metric Spaces ◽

10.1515/agms-2020-0111 ◽

2020 ◽

Vol 8 (1) ◽

pp. 114-165

Author(s):

Tetsu Toyoda

Keyword(s):

Metric Space ◽

Isometric Embedding ◽

Metric Spaces ◽

Necessary Conditions ◽

New Approach ◽

Intrinsic Characterization ◽

New Family

AbstractGromov (2001) and Sturm (2003) proved that any four points in a CAT(0) space satisfy a certain family of inequalities. We call those inequalities the ⊠-inequalities, following the notation used by Gromov. In this paper, we prove that a metric space X containing at most five points admits an isometric embedding into a CAT(0) space if and only if any four points in X satisfy the ⊠-inequalities. To prove this, we introduce a new family of necessary conditions for a metric space to admit an isometric embedding into a CAT(0) space by modifying and generalizing Gromov’s cycle conditions. Furthermore, we prove that if a metric space satisfies all those necessary conditions, then it admits an isometric embedding into a CAT(0) space. This work presents a new approach to characterizing those metric spaces that admit an isometric embedding into a CAT(0) space.

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