scholarly journals A Universal Separable Diversity

2017 ◽  
Vol 5 (1) ◽  
pp. 138-151 ◽  
Author(s):  
David Bryant ◽  
André Nies ◽  
Paul Tupper
Keyword(s):  
Metric Space ◽  
Metric Spaces ◽  
Complete Metric Space ◽  
Urysohn Space ◽  
Fundamental Properties ◽  
Finite Set ◽  
Whole Space ◽  
Set Of Points ◽  
Separable Metric Space

AbstractThe Urysohn space is a separable complete metric space with two fundamental properties: (a) universality: every separable metric space can be isometrically embedded in it; (b) ultrahomogeneity: every finite isometry between two finite subspaces can be extended to an auto-isometry of the whole space. The Urysohn space is uniquely determined up to isometry within separable metric spaces by these two properties. We introduce an analogue of the Urysohn space for diversities, a recently developed variant of the concept of a metric space. In a diversity any finite set of points is assigned a non-negative value, extending the notion of a metric which only applies to unordered pairs of points.We construct the unique separable complete diversity that it is ultrahomogeneous and universal with respect to separable diversities.

scholarly journals Duality of Moduli and Quasiconformal Mappings in Metric Spaces

2020 ◽  
Vol 8 (1) ◽  
pp. 166-181
Author(s):  
Rebekah Jones ◽  
Panu Lahti
Keyword(s):  
Metric Space ◽  
Metric Spaces ◽  
Complete Metric Space ◽  
Poincaré Inequality ◽  
Quasiconformal Mappings ◽  
Duality Relation ◽  
Doubling Measure ◽  
Poincare Inequality ◽  
Family Of Curves ◽  
The Family

AbstractWe prove a duality relation for the moduli of the family of curves connecting two sets and the family of surfaces separating the sets, in the setting of a complete metric space equipped with a doubling measure and supporting a Poincaré inequality. Then we apply this to show that quasiconformal mappings can be characterized by the fact that they quasi-preserve the modulus of certain families of surfaces.

Topological Completeness of Function Spaces Arising in the Hausdorff Approximation of Functions

1992 ◽  
Vol 35 (4) ◽  
pp. 439-448 ◽  
Author(s):  
Gerald Beer
Keyword(s):  
Approximation Theory ◽  
Metric Space ◽  
Polish Space ◽  
Complete Metric Space ◽  
Approximation Of Functions ◽  
Lower Envelopes ◽  
Constructive Approximation ◽  
Hausdorff Approximation ◽  
Set Of Points ◽  
Completely Metrizable

AbstractLet X be a complete metric space. Viewing continuous real functions on X as closed subsets of X × R, equipped with Hausdorff distance, we show that C(X, R) is completely metrizable provided X is complete and sigma compact. Following the Bulgarian school of constructive approximation theory, a bounded discontinuous function may be identified with its completed graph, the set of points between the upper and lower envelopes of the function. We show that the space of completed graphs, too, is completely metrizable, provided X is locally connected as well as sigma compact and complete. In the process, when X is a Polish space, we provide a simple answer to the following foundational question: which subsets of X × R arise as completed graphs?

scholarly journals Best Proximity Point for Generalized Proximal Weak Contractions in Complete Metric Space

2014 ◽  
Vol 2014 ◽  
pp. 1-6 ◽  
Author(s):  
Erdal Karapınar ◽  
V. Pragadeeswarar ◽  
M. Marudai
Keyword(s):  
Approximate Solution ◽  
Metric Space ◽  
Existence And Uniqueness ◽  
Metric Spaces ◽  
Best Proximity Point ◽  
Complete Metric Space ◽  
Optimal Approximate Solution ◽  
Weak Contraction ◽  
Complete Metric Spaces ◽  
New Class

We introduce a new class of nonself-mappings, generalized proximal weak contraction mappings, and prove the existence and uniqueness of best proximity point for such mappings in the context of complete metric spaces. Moreover, we state an algorithm to determine such an optimal approximate solution designed as a best proximity point. We establish also an example to illustrate our main results. Our result provides an extension of the related results in the literature.

scholarly journals Suzuki ψF-contractions and some fixed point results

2018 ◽  
Vol 34 (1) ◽  
pp. 93-102
Author(s):  
NICOLAE-ADRIAN SECELEAN ◽  
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Point Theorem ◽  
Metric Space ◽  
Metric Spaces ◽  
Complete Metric Space ◽  
Continuity Point ◽  
Picard Operator ◽  
Nonlinear Anal ◽  
New Type

The purpose of this paper is to combine and extend some recent fixed point results of Suzuki, T., [A new type of fixed point theorem in metric spaces, Nonlinear Anal., 71 (2009), 5313–5317] and Secelean, N. A. & Wardowski, D., [ψF-contractions: not necessarily nonexpansive Picard operators, Results Math., 70 (2016), 415–431]. The continuity and the completeness conditions are replaced by orbitally continuity and orbitally completeness respectively. It is given an illustrative example of a Picard operator on a non complete metric space which is neither nonexpansive nor expansive and has a unique continuity point.

On a theorem of Brian Fisher in the framework of w-distance

2017 ◽  
Vol 33 (2) ◽  
pp. 199-205
Author(s):  
DARKO KOCEV ◽  
◽  
VLADIMIR RAKOCEVIC ◽  
Keyword(s):  
Fixed Point ◽  
Metric Space ◽  
Metric Spaces ◽  
Fixed Point Theory ◽  
Complete Metric Space ◽  
Point Theory ◽  
Common Fixed Points ◽  
Complete Metric Spaces ◽  
Condition C ◽  
Relaxing Condition

In 1980. Fisher in [Fisher, B., Results on common fixed points on complete metric spaces, Glasgow Math. J., 21 (1980), 165–167] proved very interesting fixed point result for the pair of maps. In 1996. Kada, Suzuki and Takahashi introduced and studied the concept of w–distance in fixed point theory. In this paper, we generalize Fisher’s result for pair of mappings on metric space to complete metric space with w–distance. The obtained results do not require the continuity of maps, but more relaxing condition (C; k). As a corollary we obtain a result of Chatterjea.

scholarly journals Minimal arcs in metric spaces

1975 ◽  
Vol 19 (4) ◽  
pp. 426-430 ◽  
Author(s):  
S. K. Hildebrand ◽  
Harold Willis Milnes
Keyword(s):  
Fixed Points ◽  
Metric Space ◽  
Metric Spaces ◽  
Complete Metric Space ◽  
Minimal Length

In this paper we discuss the existence of an are of minimal length joining two arbitrary, yet fixed points in a complete metric space, where the metric is restricted only by the properties (A) and (B) given below. It is shown that under these conditions an arc of least length joining any two fixed points exists, and is unique. In addition, its length is shown to be equal to the metrie distance between the points.

Weak-open images of locally separable metric spaces

2011 ◽  
Vol 48 (2) ◽  
pp. 145-159
Author(s):  
Zhaowen Li ◽  
Xun Ge ◽  
Qingguo Li
Keyword(s):  
Metric Space ◽  
Metric Spaces ◽  
Weak Base ◽  
Cosmic Space ◽  
Locally Separable Metric Spaces ◽  
Separable Metric Space

In this paper, we prove that a space X is a weak-open compact image of a locally separable metric space if and only if X has a uniform cosmic-weak-base if and only if X is a weak-open compact image of a metric space and a locally cosmic space, and give some internal characterizations of weak-open s-images of locally separable metric spaces.

BISECTORS IN VECTOR GROUPS OVER INTEGERS

2019 ◽  
Vol 100 (3) ◽  
pp. 353-361 ◽  
Author(s):  
SHENG BAU ◽  
YIMING LEI
Keyword(s):  
Metric Space ◽  
Metric Spaces ◽  
Metric Dimension ◽  
Generating Set ◽  
Finite Set ◽  
Unit Vectors

We present an example of an isometric subspace of a metric space that has a greater metric dimension. We also show that the metric spaces of vector groups over the integers, defined by the generating set of unit vectors, cannot be resolved by a finite set. Bisectors in the spaces of vector groups, defined by the generating set consisting of unit vectors, are completely determined.

scholarly journals Compact reduction in Lipschitz-free spaces

2021 ◽  
Vol 151 (6) ◽  
pp. 1683-1699
Author(s):  
Ramón J. Aliaga ◽  
Camille Noûs ◽  
Colin Petitjean ◽  
Antonín Procházka
Keyword(s):  
Banach Space ◽  
Metric Space ◽  
General Principle ◽  
Metric Spaces ◽  
Approximation Property ◽  
Complete Metric Space ◽  
Infinite Dimensional ◽  
Schur Property ◽  
Free Spaces ◽  
Compact Subsets

We prove a general principle satisfied by weakly precompact sets of Lipschitz-free spaces. By this principle, certain infinite dimensional phenomena in Lipschitz-free spaces over general metric spaces may be reduced to the same phenomena in free spaces over their compact subsets. As easy consequences we derive several new and some known results. The main new results are: $\mathcal {F}(X)$ is weakly sequentially complete for every superreflexive Banach space $X$, and $\mathcal {F}(M)$ has the Schur property and the approximation property for every scattered complete metric space $M$.

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.

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