Uniqueness of spaces pretangent to metric spaces at infinity

Abstract

We find the necessary and sufficient conditions under which an unbounded metric space X has, at infinity, a unique pretangent space Ωˣ∞, ř for every scaling sequence ř. In particular, it is proved that Ωˣ∞, ř is unique and isometric to the closure of X for every logarithmic spiral X and every ř. It is also shown that the uniqueness of pretangent spaces to subsets of a real line is closely related to the “asymptotic asymmetry” of these subsets.