Spaces of metrics are Baire

IF 0.6 4区数学 Q3 MATHEMATICS

Topology and its Applications Pub Date : 2025-05-05 DOI:10.1016/j.topol.2025.109408

Yoshito Ishiki

引用次数: 0

Abstract

For a metrizable space, we consider the space of all metrics generating the same topology of the metrizable space, and this space of metrics is equipped with the supremum metric. In this paper, for every metrizable space, we establish that the space of metrics on the metrizable space is Baire. We also show that the set of all complete metrics is comeager in the space of metrics. Moreover, we investigate non–Archimedean analogues of these results. As an application, we prove that typical metrics take typical reals.

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度量空间是贝尔空间

对于一个可度量空间，我们考虑所有度量空间产生相同的可度量空间拓扑，并且该度量空间具有最高度量。对于每一个可度量空间，我们建立了可度量空间上的度量空间是贝尔空间。我们还证明了所有完备度量的集合在度量空间中是相合的。此外，我们研究了这些结果的非阿基米德类似物。作为一个应用，我们证明了典型度量取典型实数。

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.