Igor Protasov
Selectors of discrete coarse spaces

Comment.Math.Univ.Carolin. 63,2 (2022) 261-267.

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\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.

Keywords: bornology; coarse space; selector

DOI: DOI 10.14712/1213-7243.2022.012
AMS Subject Classification: 54C65