Asymptotic structure of general metric spaces at infinity

TL;DR

This paper introduces the concept of pretangent spaces at infinity for unbounded metric spaces, proving their completeness and establishing conditions for their finiteness, thereby advancing the understanding of metric space asymptotics.

Contribution

It defines pretangent spaces at infinity, proves their completeness for all unbounded metric spaces, and identifies conditions under which these spaces are finite.

Findings

Pretangent spaces are complete for all unbounded metric spaces.

Finiteness conditions for pretangent spaces are established.

Provides a framework for analyzing the asymptotic structure of metric spaces.

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.