Metric characterizations of some subsets of the real line

Abstract

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 $\ne 4$ 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 $c\in X$ and positive real number $r\in d[X^2]$, the sphere $S(c;r):=\{x\in X:d(x,c)=r\}$ 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 $G\subseteq \mathbb{R}$, a metric space $(X,d)$ is isometric to $G$ if and only if $X$ is a $2$-subline with $d[X^2]=G_{+}:=G\cap [0,\infty )$. A metric space $(X,d)$ is called a $ray$ if $X$ is a $1$-subline and $X$ contains a point $o\in X$ such that for every $r\in d[X^2]$ the sphere $S(o;r)$ is a singleton. We prove that for a subgroup $G\subseteq \mathbb{Q}$, a metric space $(X,d)$ is isometric to the ray $G_{+}$ if and only if $X$ is a ray with $d[X^2]=G_{+}$. A metric space $X$ is isometric to the ray ${\mathbb{R}}_{+}$ if and only if $X$ is a complete ray such that ${\mathbb{Q}}_{+}\subseteq d[X^2]$. On the other hand, the real line contains a dense ray $X\subseteq \mathbb{R}$ such that $d[X^2]={\mathbb{R}}_{+}$.

8 pages