2024/09/26 by Yoshito Ishiki, Ishiki, Yoshito, Katsuhisa Koshino +1
Mathematics · #Advanced Topology and Set Theory #FOS: Mathematics #Functional Analysis (math.FA) #General Topology (math.GN) #Geometric Analysis and Curvature Flows #Homotopy and Cohomology in Algebraic Topology #Metric Geometry (math.MG)
paper · pdf · doi:10.48550/arxiv.2409.17701
openalex publication_date 2024/09/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 prove that the space of bounded metrics on Z is universal for all bounded metric spaces (with restricted cardinality). Next, in contrast, we show that if Z is an infinite discrete space, then the space of metrics on Z is universal for all separable metric spaces. 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.