scholarly journals Non archimedean metric induced fuzzy uniform spaces

1989 ◽  
Vol 12 (1) ◽  
pp. 47-59 ◽  
Author(s):  
R. Lowen ◽  
A. K. Srivastava ◽  
p. Wuyts
Keyword(s):  
Metric Spaces ◽  
Uniform Spaces ◽  
Lipschitz Maps ◽  
Bireflective Subcategory

It is shown that the category of non-Archimedean metric spaces with 1-Lipschitz maps can be embedded as a coreflective non-bireflective subcategory in the category of fuzzy uniform spaces. Consequential characterizations of topological and uniform properties are derived.

Quantale-Valued Uniformizations of Quantale-Valued Generalizations of Approach Groups

2019 ◽  
Vol 15 (03) ◽  
pp. 517-538
Author(s):  
T. M. G. Ahsanullah ◽  
Gunther Jäger
Keyword(s):  
Gauge Group ◽  
Metric Spaces ◽  
Uniform Space ◽  
Uniform Spaces ◽  
Gauge Space ◽  
Bireflective Subcategory

We introduce the categories of quantale-valued approach uniform spaces and quantale-valued uniform gauge spaces, and prove that they are topological categories. We first show that the category of quantale-valued uniform gauge spaces is a full bireflective subcategory of the category of quantale-valued approach uniform spaces and; second, we prove that only under strong restrictions on the quantale these two categories are isomorphic. Besides presenting embeddings of the category of quantale-valued metric spaces into the categories of quantale-valued approach uniform spaces as well as quantale-valued uniform gauge spaces, we show that every quantale-valued approach system group and quantale-valued gauge group has a natural underlying quantale-valued approach uniform space, respectively, a quantale-valued uniform gauge space.

ON THE QUANTIFICATION OF UNIFORM PROPERTIES.PART 2

2001 ◽  
Vol 37 (1-2) ◽  
pp. 169-184
Author(s):  
B. Windels
Keyword(s):  
Metric Spaces ◽  
Uniform Spaces ◽  
Complete Metric Spaces ◽  
The Subject

In 1930 Kuratowski introduced the measure of non-compactness for complete metric spaces in order to measure the discrepancy a set may have from being compact.Since then several variants and generalizations concerning quanti .cation of topological and uniform properties have been studied.The introduction of approach uniform spaces,establishes a unifying setting which allows for a canonical quanti .cation of uniform concepts,such as completeness,which is the subject of this article.

Fuzzy Lipschitz maps and fixed point theorems in fuzzy metric spaces

2010 ◽  
Vol 161 (8) ◽  
pp. 1117-1130 ◽  
Author(s):  
Gabjin Yun ◽  
Seungsu Hwang ◽  
Jeongwook Chang
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorems ◽  
Metric Spaces ◽  
Fuzzy Metric Spaces ◽  
Fuzzy Metric ◽  
Lipschitz Maps

scholarly journals Lipschitz Extensions to Finitely Many Points

2018 ◽  
Vol 6 (1) ◽  
pp. 174-191 ◽  
Author(s):  
Giuliano Basso
Keyword(s):  
Banach Space ◽  
Hilbert Space ◽  
Metric Spaces ◽  
Lipschitz Constant ◽  
Target Space ◽  
Square Root ◽  
Lipschitz Maps ◽  
Lipschitz Map ◽  
Lipschitz Extensions

AbstractWe consider Lipschitz maps with values in quasi-metric spaces and extend such maps to finitely many points. We prove that in this context every 1-Lipschitz map admits an extension such that its Lipschitz constant is bounded from above by the number of added points plus one. Moreover, we prove that if the source space is a Hilbert space and the target space is a Banach space, then there exists an extension such that its Lipschitz constant is bounded from above by the square root of the total of added points plus one. We discuss applications to metric transforms.

scholarly journals Dynamic Processes, Fixed Points, Endpoints, Asymmetric Structures, and Investigations Related to Caristi, Nadler, and Banach in Uniform Spaces

2015 ◽  
Vol 2015 ◽  
pp. 1-16 ◽  
Author(s):  
Kazimierz Włodarczyk ◽  
Robert Plebaniak
Keyword(s):  
Fixed Point ◽  
Fixed Points ◽  
Dynamic Systems ◽  
Metric Spaces ◽  
Lower Semicontinuous ◽  
Type Theorem ◽  
Uniform Spaces ◽  
Asymmetric Structures ◽  
Symmetric Structures ◽  
Locally Convex

In uniform spacesX, Dwith symmetric structures determined by theD-families of pseudometrics which define uniformity in these spaces, the new symmetric and asymmetric structures determined by theJ-families of generalized pseudodistances onXare constructed; using these structures the set-valued contractions of two kinds of Nadler type are defined and the new and general theorems concerning the existence of fixed points and endpoints for such contractions are proved. Moreover, using these new structures, the single-valued contractions of two kinds of Banach type are defined and the new and general versions of the Banach uniqueness and iterate approximation of fixed point theorem for uniform spaces are established. Contractions defined and studied here are not necessarily continuous. One of the main key ideas in this paper is the application of our fixed point and endpoint version of Caristi type theorem for dissipative set-valued dynamic systems without lower semicontinuous entropies in uniform spaces with structures determined byJ-families. Results are new also in locally convex and metric spaces. Examples are provided.

scholarly journals Quantifying completion

2000 ◽  
Vol 23 (11) ◽  
pp. 729-739 ◽  
Author(s):  
Robert Lowen ◽  
Bart Windels
Keyword(s):  
Metric Spaces ◽  
Uniform Spaces ◽  
Approach Spaces

Approach uniformities were introduced in Lowen and Windels (1998) as the canonical generalization of both metric spaces and uniform spaces. This text presents in this new context of “quantitative” uniform spaces, a reflective completion theory which generalizes the well-known completions of metric and uniform spaces. This completion behaves nicely with respect to initial structures and hyperspaces. Also, continuous extensions of pseudo-metrics on uniform spaces and (real) compactification of approach spaces can be interpreted in terms of this completion.

scholarly journals Coupled fixed point results on metric spaces defined by binary operations

2018 ◽  
Vol 10 (2) ◽  
pp. 313-323
Author(s):  
A. Karami ◽  
R. Shakeri ◽  
S. Sedghi ◽  
І. Altun
Keyword(s):  
Fixed Point ◽  
Metric Space ◽  
Metric Spaces ◽  
Coupled Fixed Point ◽  
Banach Fixed Point Theorem ◽  
Partial Metric ◽  
Uniform Spaces ◽  
Binary Operations ◽  
Fundamental Properties ◽  
Partial Metric Spaces

In parallel with the various generalizations of the Banach fixed point theorem in metric spaces, this theory is also transported to some different types of spaces including ultra metric spaces, fuzzy metric spaces, uniform spaces, partial metric spaces, $b$-metric spaces etc. In this context, first we define a binary normed operation on nonnegative real numbers and give some examples. Then we recall the concept of $T$-metric space and some important and fundamental properties of it. A $T$-metric space is a $3$-tuple $(X, T, \diamond)$, where $X$ is a nonempty set, $\diamond$ is a binary normed operation and $T$ is a $T$-metric on $X$. Since the triangular inequality of $T$-metric depends on a binary operation, which includes the sum as a special case, a $T$-metric space is a real generalization of ordinary metric space. As main results, we present three coupled fixed point theorems for bivariate mappings satisfying some certain contractive inequalities on a complete $T$-metric space. It is easily seen that not only existence but also uniqueness of coupled fixed point guaranteed in these theorems. Also, we provide some suitable examples that illustrate our results.

A generalization of isometries to uniform spaces

1956 ◽  
Vol 52 (3) ◽  
pp. 399-405 ◽  
Author(s):  
F. Rhodes
Keyword(s):  
Metric Spaces ◽  
Uniform Space ◽  
Compact Groups ◽  
Uniform Spaces ◽  
Totally Bounded ◽  
Transitive Groups ◽  
Groups Of Automorphisms

Isomorphisms are, in many ways, the generalizations of isometrics to uniform spaces. Yet some theorems on isometries of metric spaces only generalize to uniform spaces in terms of more restricted transformations of the uniform space. In § 1, in the course of a discussion of a theorem on transitive groups of automorphisms, we define such a transformation and call it an isobasism. It appears that in many respects isobasisms, rather than isomorphisms, are the generalizations of isometries to uniform spaces. The results of Freudenthal and Hurewicz (7) on contractions, expansions and isometries of totally bounded metric spaces are generalized, in § 2, to contractions, expansions and isobasisms of totally bounded uniform spaces. These results, together with generalizations of some theorems of Eilenberg (6) on compact groups of homeomorphisms of metric spaces which are obtained in §3, give a characterization of isobasisms. The language of Bourbaki (2,3,4) is used throughout this note.

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