# Metric characterizations of some subsets of the real line

• I. Banakh Institute for Applied Problems of Mechanics and Mathematics of National Academy of Sciences of Ukraine, Lviv, Ukraine
• T. Banakh Ivan Franko National University of Lviv
• M. Kolinko Ivan Franko National University of Lviv, Lviv, Ukraine
• A. Ravsky Institute for Applied Problems of Mechanics and Mathematics of National Academy of Sciences of Ukraine, Lviv, Ukraine
Keywords: metric space; triangle equality; subline; ray

### Abstract

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

### Author Biographies

I. Banakh, Institute for Applied Problems of Mechanics and Mathematics of National Academy of Sciences of Ukraine, Lviv, Ukraine

Institute for Applied Problems of Mechanics and Mathematics of National Academy of Sciences of Ukraine, Lviv, Ukraine

T. Banakh, Ivan Franko National University of Lviv

Ivan Franko National University of Lviv

M. Kolinko, Ivan Franko National University of Lviv, Lviv, Ukraine

Ivan Franko National University of Lviv, Lviv, Ukraine

A. Ravsky, Institute for Applied Problems of Mechanics and Mathematics of National Academy of Sciences of Ukraine, Lviv, Ukraine

Institute for Applied Problems of Mechanics and Mathematics of National Academy of Sciences of Ukraine, Lviv, Ukraine

### References

T. Banakh, Banakh spaces and their geometry, arxiv.org/abs/2305.07354.

I. Banakh, T. Banakh, M. Kolinko, A. Ravsky, Midconvex sets in Abelian groups, arxiv.org/abs/ 2305.12128.

I. Banakh, T. Banakh, M. Kolinko, A. Ravsky, Semiaffine sets in Abelian groups, arxiv.org/abs/ 2305.07905.

J. Bowers, P. Bowers, A Menger redux: embedding metric spaces isometrically in Euclidean space, Amer. Math. Monthly, 124 (2017), №7, 621–636.

W. Brian, https://mathoverflow.net/a/442872/61536.

R. Engelking, General topology, Heldermann Verlag, Berlin, 1989.

K. Menger, New foundation of Euclidean geometry, Amer. J. Math., 53 (1931), №4, 721–745.

D. Robinson, A course in the theory of groups, Springer-Verlag, New York, 1996.

Published
2023-06-23
How to Cite
Banakh, I., Banakh, T., Kolinko, M., & Ravsky, A. (2023). Metric characterizations of some subsets of the real line. Matematychni Studii, 59(2), 205-214. https://doi.org/10.30970/ms.59.2.205-214
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Articles