Selectors of discrete coarse spaces

Protasov, Igor. "Selectors of discrete coarse spaces." Commentationes Mathematicae Universitatis Carolinae 62 63.2 (2022): 261-267. <http://eudml.org/doc/298524>.

@article{Protasov2022,
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 “$\le $" on $X$ such that the family of intervals $\lbrace [a, b]\colon a,b\in X, a\le b \rbrace $ is a base for the bornology $\mathcal \{B\}$.},
author = {Protasov, Igor},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {bornology; coarse space; selector},
language = {eng},
number = {2},
pages = {261-267},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Selectors of discrete coarse spaces},
url = {http://eudml.org/doc/298524},
volume = {62 63},
year = {2022},
}

TY - JOUR
AU - Protasov, Igor
TI - Selectors of discrete coarse spaces
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2022
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 62 63
IS - 2
SP - 261
EP - 267
AB - 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 “$\le $" on $X$ such that the family of intervals $\lbrace [a, b]\colon a,b\in X, a\le b \rbrace $ is a base for the bornology $\mathcal {B}$.
LA - eng
KW - bornology; coarse space; selector
UR - http://eudml.org/doc/298524
ER -