Selectors of discrete coarse spaces

TL;DR

This paper characterizes when discrete coarse spaces admit selectors and 2-selectors, linking these properties to the existence of a linear order whose intervals form a base for the space's bornology.

Contribution

It establishes an equivalence between the existence of selectors in discrete coarse spaces and the presence of a compatible linear order with a specific interval base.

Findings

Selectors exist if and only if a compatible linear order exists.

Selectors are equivalent to 2-selectors in discrete coarse spaces.

The interval family from the linear order forms a base for the bornology.

Abstract

Given a coarse space $(X, \mathcal{E})$ with the bornology $\mathcal B$ of bounded subsets, we extend the coarse structure $\mathcal E$ from $X\times X$ to the natural coarse structure on $(\mathcal B \backslash \lbrace \emptyset\rbrace)\times (\mathcal B \backslash \lbrace \emptyset\rbrace)$ and say that a macro-uniform mapping $f: (\mathcal B \backslash \lbrace \emptyset\rbrace)\rightarrow X$ (resp. $f: [ X]^2 \rightarrow X$) is a selector (resp. 2-selector) of $(X, \mathcal{E})$ if $f(A)\in A$ for each $A\in \mathcal B\setminus \lbrace\emptyset\rbrace$ (resp. $A \in [X]^2 )$. We prove that a discrete coarse space $(X, \mathcal{E})$ admits a selector if and only if $(X, \mathcal{E})$ admits a 2-selector if and only if there exists a linear order $\leq$ on $X$ such that the family of intervals $\lbrace [a, b]: a,b\in X, \ a\leq b \}$ is a base for the bornology $\mathcal B$.