Metric characterizations of some subsets of the real line
Abstract
A metric space (X,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 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 {\em 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 {\em 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+.
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