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.

37 pages, 3 figures

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