Asymptotic structure of general metric spaces at infinity

Abstract

Let $(X,d)$ be an unbounded metric space and $\tilde r=(r_n)_{n\in\mathbb N}$ be a scaling sequence of positive real numbers tending to infinity. We define the pretangent space $\Omega_{\infty, \tilde r}^{X}$ to $(X, d)$ at infinity as a metric space whose points are equivalence classes of sequences $(x_n)_{n\in\mathbb N}\subset X$ which tend to infinity with the speed of $\tilde r$. It is proved that the pretangent spaces $\Omega_{\infty, \tilde r}^{X}$ are complete for every unbounded metric space $(X, d)$ and every scaling sequence $\tilde r$. The finiteness conditions of $\Omega_{\infty, \tilde r}^{X}$ are found.