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

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.

scholarly journals The isometric embedding problem for length metric spaces

2019 ◽  
pp. 1-44
Author(s):  
Barry Minemyer
Keyword(s):  
Metric Space ◽  
Isometric Embedding ◽  
Metric Spaces ◽  
Energy Functional ◽  
Embedding Problem ◽  
Lorentzian Space ◽  
Set Of Points ◽  
Dense Set

We prove that every proper [Formula: see text]-dimensional length metric space admits an “approximate isometric embedding” into Lorentzian space [Formula: see text]. By an “approximate isometric embedding” we mean an embedding which preserves the energy functional on a prescribed set of geodesics connecting a dense set of points.

scholarly journals Closedness, separation and connectedness in pseudo-quasi-semi metric spaces

Filomat ◽  
2020 ◽  
Vol 34 (14) ◽  
pp. 4757-4766
Author(s):  
Tesnim Baran
Keyword(s):  
Metric Space ◽  
Metric Spaces ◽  
Closure Operator ◽  
Topological Category ◽  
The Relationship

In this paper, we give the characterization of closed and strongly closed subsets of an extended pseudo-quasi-semi metric space and show that they induce closure operator. Moreover, we characterize each of Ti, i = 0, 1, 2 and connected extended pseudo-quasi-semi metric spaces and investigate the relationship among them. Finally, we introduce the notion of irreducible objects in a topological category and examine the relationship among each of irreducible, Ti,i = 1,2, and connected extended pseudo-quasi-semi metric spaces.

scholarly journals Completeness of b∼Metric Spaces and the Fixed Points of Generalized Multivalued Quasicontractions

2020 ◽  
Vol 2020 ◽  
pp. 1-13
Author(s):  
Basit Ali ◽  
Mujahid Abbas ◽  
Manuel de la Sen
Keyword(s):  
Fixed Points ◽  
Metric Space ◽  
Topological Property ◽  
Metric Spaces

In this article, we present a completeness characterization of b∼metric space via existence of fixed points of generalized multivalued quasicontractions. The purpose of this paper is twofold: (a) to establish the existence of fixed points of multivalued quasicontractions in the setup of b∼ metric spaces and (b) to establish completeness of a b∼ metric space which is a topological property in nature with existence of fixed points of generalized multivalued quasicontractions. Further, a comparison of our results with comparable results shows that the results obtained herein improve and unify the existing results in the literature applicable to the case where existing results fail.

scholarly journals A Characterization of Strong Completeness in Fuzzy Metric Spaces

Mathematics ◽  
2020 ◽  
Vol 8 (6) ◽  
pp. 861
Author(s):  
Valentín Gregori ◽  
Juan-José Miñana ◽  
Bernardino Roig ◽  
Almanzor Sapena
Keyword(s):  
Metric Space ◽  
Metric Spaces ◽  
Fixed Point Theory ◽  
Point Theory ◽  
Fuzzy Metric Space ◽  
Strong Completeness ◽  
Fuzzy Metric ◽  
Fuzzy Fixed Point ◽  
Nested Sequence

Here, we deal with the concept of fuzzy metric space ( X , M , ∗ ) , due to George and Veeramani. Based on the fuzzy diameter for a subset of X , we introduce the notion of strong fuzzy diameter zero for a family of subsets. Then, we characterize nested sequences of subsets having strong fuzzy diameter zero using their fuzzy diameter. Examples of sequences of subsets which do or do not have strong fuzzy diameter zero are provided. Our main result is the following characterization: a fuzzy metric space is strongly complete if and only if every nested sequence of close subsets which has strong fuzzy diameter zero has a singleton intersection. Moreover, the standard fuzzy metric is studied as a particular case. Finally, this work points out a route of research in fuzzy fixed point theory.

Distance Sets of Urysohn Metric Spaces

2013 ◽  
Vol 65 (1) ◽  
pp. 222-240 ◽  
Author(s):  
N.W. Sauer
Keyword(s):  
Metric Space ◽  
Metric Spaces ◽  
Finite Metric Space ◽  
Distance Set ◽  
Distance Sets ◽  
Separable Metric Space

Abstract.A metric space M = (M; d) is homogeneous if for every isometry f of a finite subspace of M to a subspace of M there exists an isometry of M onto M extending f . The space M is universal if it isometrically embeds every finite metric space F with dist(F) ⊆ dist(M) (with dist(M) being the set of distances between points in M).A metric space U is a Urysohn metric space if it is homogeneous, universal, separable, and complete. (We deduce as a corollary that a Urysohn metric space U isometrically embeds every separable metric space M with dist(M) ⊆ dist(U).)The main results are: (1) A characterization of the sets dist(U) for Urysohn metric spaces U. (2) If R is the distance set of a Urysohn metric space and M and N are two metric spaces, of any cardinality with distances in R, then they amalgamate disjointly to a metric space with distances in R. (3) The completion of every homogeneous, universal, separable metric space M is homogeneous.

scholarly journals PTOLEMAIC SPACES AND CAT(0)

2009 ◽  
Vol 51 (2) ◽  
pp. 301-314 ◽  
Author(s):  
S. M. BUCKLEY ◽  
K. FALK ◽  
D. J. WRAITH
Keyword(s):  
Euclidean Space ◽  
Riemannian Manifolds ◽  
Metric Space ◽  
Metric Spaces ◽  
Finsler Manifold ◽  
Full Generality ◽  
Space Setting ◽  
Ptolemaic Space

AbstractWe consider Ptolemy's inequality in a metric space setting. It is not hard to see that CAT(0) spaces satisfy this inequality. Although the converse is not true in full generality, we show that if our Ptolemaic space is either a Riemannian or Finsler manifold, then it must also be CAT(0). Ptolemy's inequality is closely related to inversions of metric spaces. We exploit this link to establish a new characterization of Euclidean space amongst all Riemannian manifolds.

A note on compact-valued continuous relations on metric spaces

2009 ◽  
Vol 46 (2) ◽  
pp. 149-156
Author(s):  
Xun Ge ◽  
Ying Ge
Keyword(s):  
Metric Space ◽  
Metric Spaces

In this paper, we give a characterization of compact-valued continuous relations on metric spaces. By this characterization, we prove that for two relations f and g on a metric space X , the composition gf of f with g is compact-valued continuous if both f and g are compact-valued continuous. As a corollary of this result, for a relation f on a metric space X , fn is a compact-valued continuous for all n ∈ ℕ iff f is a compact-valued continuous, which improves a result of H. Y. Chu and J. S. Park by omitting locally compactness of X .

scholarly journals Fractal Sets as Final Coalgebras Obtained by Completing an Initial Algebra

10.29007/pw5g ◽  
2018 ◽  
Author(s):  
Larry Moss ◽  
Jayampathy Ratnayake ◽  
Robert Rose
Keyword(s):  
Metric Space ◽  
Metric Spaces ◽  
Topological Spaces ◽  
Fractal Sets ◽  
Cauchy Completion ◽  
Sweeping Generalization ◽  
Initial Algebra ◽  
Finite Metric Spaces ◽  
Set Of Points

This paper is a contribution to the presentation of fractal sets in terms of final coalgebras.The first result on this topic was Freyd's Theorem: the unit interval [0,1] is the final coalgebra ofa certain functor on the category of bipointed sets. Leinster 2011 offersa sweeping generalization of this result. He is able to represent many of what would be intuitivelycalled "self-similar" spaces using (a) bimodules (also called profunctors or distributors),(b) an examination of non-degeneracy conditions on functors of various sorts; (c) a construction offinal coalgebras for the types of functors of interest using a notion of resolution. In addition to thecharacterization of fractals sets as sets, his seminal paper also characterizes them as topological spaces.Our major contribution is to suggest that in many cases of interest, point (c) above on resolutionsis not needed in the construction of final coalgebras. Instead, one may obtain a number of spaces ofinterest as the Cauchy completion of an initial algebra,and this initial algebra is the set of points in a colimit of an omega-sequence of finite metric spaces.This generalizes Hutchinson's 1981 characterization of fractal attractors asclosures of the orbits of the critical points. In addition to simplifying the overall machinery, it also presents a metric space which is ``computationally related'' to the overall fractal. For example, when applied to Freyd's construction, our method yields the metric space.of dyadic rational numbers in [0,1].Our second contribution is not completed at this time, but it is a set of results on \emph{metric space}characterizations of final coalgebras. This point was raised as an open issue in Hasuo, Jacobs, and Niqui 2010,and our interest in quotient metrics comes from their paper. So in terms of (a)--(c) above, our workdevelops (a) and (b) in metric settings while dropping (c).

A fuzzy fixed point problem for constraint inequalities

2020 ◽  
Vol 39 (3) ◽  
pp. 3025-3032
Author(s):  
Hüseyin Işık ◽  
Muzeyyen Sangurlu Sezen
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Metric Space ◽  
Metric Spaces ◽  
Fixed Point Problem ◽  
Point Problem ◽  
New Approach ◽  
Fuzzy Metric ◽  
Constraint Inequalities ◽  
Fuzzy Fixed Point

In this work, we prove a new fixed point theorem in the setting fuzzy metric spaces. The fuzzy metric space considered here is assumed to have two partial orders defined on it. We introduce a new approach to the existence of a fixed point of a function satisfying the two constraint inequalities. An example is included which illustrates new results of this paper. Moreover, an application of our result to the study of integral equations is provided.

A Characterization of 2-Betweenness in 2-Metric Spaces

1966 ◽  
Vol 18 ◽  
pp. 963-968 ◽  
Author(s):  
Raymond W. Freese ◽  
Edward Z. Andalafte
Keyword(s):  
Metric Space ◽  
Metric Spaces ◽  
Geometric Properties ◽  
Natural Properties ◽  
Similar Theorem ◽  
Area Metric ◽  
Initial Work ◽  
Object Of Study

The topology of abstract 2-metric (area-metric) spaces has been the object of study in recent papers of Gähler (1) and Froda (2). The geometric properties of such spaces, however, have remained largely untouched since the initial work of Menger (3). As in ordinary metric spaces, a notion of 2-betweenness, or interiorness, can be easily defined in 2-metric spaces. In abstract metric spaces the betweenness relation is characterized among all relations defined on each triple of points of every metric space by six natural properties (4, pp. 33-40; 5). The purpose of this paper is to prove a similar theorem characterizing the relation of 2-betweenness in 2-metric spaces.

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