2019/03/22 by Mahmood Etedadialiabadi, Etedadialiabadi, Mahmood, Gao, Su · 1 citation
Decision Sciences · Mathematics · #05C12 #51F99 #Advanced Banach Space Theory #Advanced Topology and Set Theory #FOS: Mathematics #Functional Analysis (math.FA) #Fuzzy and Soft Set Theory #Logic (math.LO) #Metric Geometry (math.MG) #Primary 05B25 #Secondary 03C13
paper · pdf · doi:10.48550/arxiv.1903.09723
openalex publication_date 2019/03/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this paper we define a notion of S-extension for a metric space and study minimality and coherence of S-extensions. We show that every S-extension can be identified with an algebraic object. We use this algebraic representation to give a complete characterization of all finite minimal S-extensions of a given finite metric space and a complete characterization of all minimal coherent S-extensions. We also define a notion of ultraextensive metric spaces and show that every countable metric space can be extended to a countable ultraextensive metric space. %As an application, we show that every countable subset of the Urysohn metric space can be extended to a countable dense ultraextensive subset of the Urysohn space. We also show that the isometry group of an infinite ultraextensive metric space has a dense locally finite subgroup, generalizing several previously known results. We also study compact ultrametric spaces and show that every compact ultrametric space can be extended to a compact ultraextensive ultrametric space.