Extension theorems for large scale spaces via coarse neighbourhoods

TL;DR

This paper extends classical topological theorems like Urysohn's and Tietze's to large scale spaces using coarse geometry, introducing hybrid large scale normal spaces and analyzing their properties.

Contribution

It introduces hybrid large scale normal spaces, proves coarse analogues of classical theorems, and characterizes their properties in metric and locally compact abelian groups.

Findings

All metric spaces are hybrid large scale normal.

Characterization of hybrid large scale normal locally compact abelian groups.

Analysis of Higson compactifications and coronas for these spaces.

Abstract

We introduce the notion of (hybrid) large scale normal space and prove coarse geometric analogues of Urysohn's Lemma and the Tietze Extension Theorem for these spaces, where continuous maps are replaced by (continuous and) slowly oscillating maps. To do so, we first prove a general form of each of these results in the context of a set equipped with a neighbourhood operator satisfying certain axioms, from which we obtain both the classical topological results and the (hybrid) large scale results as corollaries. We prove that all metric spaces are hybrid large scale normal, and characterize those locally compact abelian groups which (as hybrid large scale spaces) are hybrid large scale normal. Finally, we look at some properties of the Higson compactifications and coronas of hybrid large scale normal spaces.