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IF 0.6 4区数学 Q3 MATHEMATICS

Topology and its Applications Pub Date : 2025-04-17 DOI:10.1016/j.topol.2025.109394

Yoshito Ishiki , Katsuhisa Koshino

引用次数: 0

摘要

如果一类度量空间中的所有度量空间都能等距嵌入到（M,d）中，则称该类度量空间（M,d）是通用的。在本文中，对于一个拥有丰富子空间的可元空间 Z，我们首先研究了 Z 上度量空间的普适性。接下来，我们反过来证明，如果 Z 是一个无限离散空间，那么 Z 上度量空间对于所有与 Z 具有相同权重的度量空间都是普适的。作为我们结果的一个推论，如果 Z 是非紧凑的，或者是不可数且紧凑的，那么 Z 上度量空间对于所有紧凑度量空间都是普适的。此外，如果 Z 是紧凑的可数空间，那么存在一个紧凑的度量空间，它不能等距嵌入到 Z 的度量空间中。

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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On isometric universality of spaces of metrics

A metric space

(M, d)

is said to be universal for a class of metric spaces if all metric spaces in the class can be isometrically embedded into

(M, d)

. In this paper, for a metrizable space Z possessing abundant subspaces, we first investigate the universality of the space of metrics on Z. Next, in contrast, we show that if Z is an infinite discrete space, then the space of metrics on Z is universal for all metric spaces having the same weight of Z. As a corollary of our results, if Z is non-compact, or uncountable and compact, then the space of metrics on Z is universal for all compact metric spaces. In addition, if Z is compact and countable, then there exists a compact metric space that can not be isometrically embedded into the space of metrics on Z.

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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.