Discreteness and Large Scale Surjections

TL;DR

This paper explores coarse disjointness and large scale $n$-to-$1$ functions, providing new characterizations of asymptotic dimension and invariance properties, and examining metrizability of large scale structures.

Contribution

It introduces an Ostrand-type characterization of asymptotic dimension and shows invariance of several properties under coarsely $n$-to-$1$ functions.

Findings

Finite asymptotic dimension is invariant under coarsely $n$-to-$1$ functions.

Coarse finitism and large scale weak paracompactness are invariants of coarsely $n$-to-$1$ functions.

Metrizability of large scale structures is also studied.

Abstract

We study the concept of coarse disjointness and large scale $n$-to-$1$ functions. As a byproduct, we obtain an Ostrand-type characterization of asymptotic dimension for coarse structures. It is shown that properties like finite asymptotic dimension, coarse finitism, large scale weak paracompactness, ect. are all invariants of coarsely $n$-to-$1$ functions. Metrizability of large scale structures is also investigated.