Finite Spaces Pretangent to Metric Spaces at Infinity

Abstract

Let X be an unbounded metric space, and let \( \tilde{r} \) be a sequence of positive real numbers tending to infinity. We define the pretangent space \( {\Omega}_{\infty, \tilde{r}}^X \) to X at infinity as a metric space whose points are the equivalence classes of sequences \( \tilde{x}\subset X \) which tend to infinity with the rate \( \tilde{r} \). It is proved that all pretangent spaces are complete and, for every finite metric space Y, there is an unbounded metric space X such that Y and \( {\Omega}_{\infty, \tilde{r}}^X \) are isometric for some \( \tilde{r} \). The finiteness conditions of \( {\Omega}_{\infty, \tilde{r}}^X \) are completely described.