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 \( {\Omega}_{\infty, \tilde{r}}^X \) for every scaling sequence \( \tilde{r} \). In particular, it is proved that \( {\Omega}_{\infty, \tilde{r}}^X \) is unique and isometric to the closure of X for every logarithmic spiral X and every \( \tilde{r} \). 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.