On equivalence of unbounded metric spaces at infinity

Abstract

Let (X, d) be an unbounded metric space. To investigate the asymptotic behavior of (X, d) at infinity, one can consider a sequence of rescaling metric spaces (X, 1 rn d) generated by given sequence (r n ) n∈N of positive reals with r n → ∞. Metric spaces that are limit points of the sequence (X, 1 rn d) n∈N will be called pretangent spaces to (X, d) at infinity. We found the necessary and sufficient conditions under which two given unbounded subspaces of (X, d) have the same pretangent spaces at infinity. In the case when (X, d) is the real line with Euclidean metric, we also describe all unbounded subspaces of (X, d) isometric to their pretangent spaces.