METRIC CHARACTERIZATIONS OF SOME SUBSETS OF THE REAL LINE.

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 ⊆ 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 ⊆ 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 R+ if and only if X is a complete ray such that Q+ ⊆ d[X2]. On the other hand, the real line contains a dense ray X ⊆ R such that d[X2] = R+. [ABSTRACT FROM AUTHOR]