Extension of unbounded uniformly continuous functions and pseudometrics

IF 0.6 4区数学 Q3 MATHEMATICS Topology and its Applications Pub Date : 2024-05-22 DOI:10.1016/j.topol.2024.108959

Michal Hevessy

引用次数: 0

Abstract

In this paper we aim to characterize uniformly continuous real functions and pseudometrics on metric spaces, having uniformly continuous extension. For functions we use a very similar approach as McShane in [7] using moduli of continuity. By doing that we obtain an explicit formula for the extension. We also show that our characterization for functions is equivalent to one proposed in [8] for uniform spaces. We then show that a similar approach can be done for uniformly continuous pseudometrics.

To do so we use the notion of chainable metric spaces and intrinsic metrics defined in [9]. A somewhat similar approach has been studied in [6] for normed linear spaces.

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无界均匀连续函数的扩展和伪计量学

本文旨在描述具有均匀连续外延的度量空间上的均匀连续实函数和伪几何。对于函数，我们采用了与 McShane 在 [7] 中非常相似的方法，即使用连续性模量。通过这种方法，我们得到了扩展的明确公式。我们还证明，我们对函数的描述等同于[8]中对均匀空间的描述。为此，我们使用了 [9] 中定义的可链度量空间和本征度量的概念。对于规范线性空间，[6] 也研究过类似的方法。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Topology and its Applications 数学-数学

CiteScore

1.20

自引率

33.30%

发文量

251

审稿时长

6 months

期刊介绍： Topology and its Applications is primarily concerned with publishing original research papers of moderate length. However, a limited number of carefully selected survey or expository papers are also included. The mathematical focus of the journal is that suggested by the title: Research in Topology. It is felt that it is inadvisable to attempt a definitive description of topology as understood for this journal. Certainly the subject includes the algebraic, general, geometric, and set-theoretic facets of topology as well as areas of interactions between topology and other mathematical disciplines, e.g. topological algebra, topological dynamics, functional analysis, category theory. Since the roles of various aspects of topology continue to change, the non-specific delineation of topics serves to reflect the current state of research in topology. At regular intervals, the journal publishes a section entitled Open Problems in Topology, edited by J. van Mill and G.M. Reed. This is a status report on the 1100 problems listed in the book of the same name published by North-Holland in 1990, edited by van Mill and Reed.

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