Discreteness and Large Scale Surjections

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.