2023/05/13 by Iryna Banakh, Banakh, Iryna, Тарас Банах +5
Mathematics · #51F99 #54E35 #54E40 #Advanced Differential Equations and Dynamical Systems #Advanced Topology and Set Theory #FOS: Mathematics #General Topology (math.GN) #Group Theory (math.GR) #Metric Geometry (math.MG) #Rings, Modules, and Algebras
paper · pdf · doi:10.48550/arxiv.2305.07907
openalex publication_date 2023/05/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
A metric space (X,d) is called a subline if every 3-element subset T of X can be written as T=\x,y,z\ for some points x,y,z such that d(x,z)=d(x,y)+d(y,z). By a classical result of Menger, every subline of cardinality ≠ 4 is isometric to a subspace of the real line. A subline (X,d) is called an n-subline for a natural number n if for every c∈ X and positive real number r∈ d[X2], the sphere S(c;r):=\x∈ X:d(x,c)=r\ contains at least n points. We prove that every 2-subline is isometric to some additive subgroup of the real line. Moreover, for every subgroup G⊆\mathbb R, a metric space (X,d) is isometric to G if and only if X is a 2-subline with d[X2]=G+:= G∩[0,∞). A metric space (X,d) is called a ray if X is a 1-subline and X contains a point o∈ X such that for every r∈ d[X2] the sphere S(o;r) is a singleton. We prove that for a subgroup G⊆\mathbb Q, a metric space (X,d) is isometric to the ray G+ if and only if X is a ray with d[X2]=G+. A metric space X is isometric to the ray \mathbb R+ if and only if X is a complete ray such that \mathbb Q+⊆ d[X2]. On the other hand, the real line contains a dense ray X⊆\mathbb R such that d[X2]=\mathbb R+.