Selectors of discrete coarse spaces

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 $(\mathcal{B}\backslash \{\varnothing \})\times (\mathcal{B}\backslash \{\varnothing \})$ and say that a macro-uniform mapping $f:(\mathcal{B}\backslash \{\varnothing \})\to X$ (resp. $f:[X{]}^2\to X$) is a selector (resp. 2-selector) of $(X,\mathcal{E})$ if $f(A)\in A$ for each $A\in \mathcal{B}\setminus \{\varnothing \}$ (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 $\le$ on $X$ such that the family of intervals $\{[a,b]:a,b\in X,a\le b\}$ is a base for the bornology $\mathcal{B}$.

bornology, coarse space, selector