Selectors of discrete coarse spaces

Protasov, Igor

Commentationes Mathematicae Universitatis Carolinae, Tome 63 (2022) no. 2, pp. 261-267.

Résumé

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 $f\colon (\mathcal B \backslash \lbrace \emptyset\rbrace)\rightarrow X$ (or $f\colon [ X]^2 \rightarrow X$) is a selector (or 2-selector) of $(X, \mathcal{E})$ if $f(A)\in A$ for each $A\in \mathcal B\setminus \lbrace\emptyset\rbrace$ ($A \in [X]^2 $, respectively). 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 $\lbrace [a, b]\colon a,b\in X, a\leq b \}$ is a base for the bornology $\mathcal B$.

Détail

DOI : 10.14712/1213-7243.2022.012

Classification : 54C65
Keywords: bornology; coarse space; selector