Article
| Title: | Selectors of discrete coarse spaces (English) |
| Author: | Protasov, Igor |
| Language: | English |
| Journal: | Commentationes Mathematicae Universitatis Carolinae |
| ISSN: | 0010-2628 (print) |
| ISSN: | 1213-7243 (online) |
| Volume: | 63 |
| Issue: | 2 |
| Year: | 2022 |
| Pages: | 261-267 |
| Summary lang: | English |
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| Category: | math |
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| Summary: | 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 $\mathcal B$. (English) |
| Keyword: | bornology |
| Keyword: | coarse space |
| Keyword: | selector |
| MSC: | 54C65 |
| idZBL: | Zbl 07613034 |
| idMR: | MR4506136 |
| DOI: | 10.14712/1213-7243.2022.012 |
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| Date available: | 2022-11-02T09:22:04Z |
| Last updated: | 2023-03-13 |
| Stable URL: | http://hdl.handle.net/10338.dmlcz/151089 |
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