| 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 |
| . |
| Category:
|
math |
| . |
| 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 |
| . |
| Date available:
|
2022-11-02T09:22:04Z |
| Last updated:
|
2024-07-01 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/151089 |
| . |
| Reference:
|
[1] Artico G., Marconi U., Pelant J., Rotter L., Tkachenko M.: Selections and suborderability.Fund. Math. 175 (2002), no. 1, 1–33. Zbl 1019.54014, MR 1971236 |
| Reference:
|
[2] Dikranjan D., Protasov I., Protasova K., Zava N.: Balleans, hyperballeans and ideals.Appl. Gen. Topology 2 (2019), 431–447. MR 4019581, 10.4995/agt.2019.11645 |
| Reference:
|
[3] Dikranjan D., Protasov I., Zava N.: Hyperballeans of groups.Topology Appl. 263 (2019), 172–198. MR 3961119, 10.1016/j.topol.2019.05.019 |
| Reference:
|
[4] Engelking R., Health R. W., Michael E.: Topological well-ordering and continuous selections.Invent. Math. 6 (1968), 150–158. MR 0244959, 10.1007/BF01425452 |
| Reference:
|
[5] de la Harpe P.: Topics in Geometric Group Theory.Chicago Lectures in Mathematics, The University Chicago Press, Chicago, 2000. MR 1786869 |
| Reference:
|
[6] van Mill J., Pelant J., Pol R.: Selections that characterize topological completeness.Fund. Math. 149 (1996), no. 2, 127–141. MR 1376668, 10.4064/fm-149-2-127-141 |
| Reference:
|
[7] van Mill J., Wattel E.: Selections and orderability.Proc. Amer. Math. Soc. 83 (1981), no. 3, 601–605. MR 0627702, 10.1090/S0002-9939-1981-0627702-4 |
| Reference:
|
[8] Protasov I., Banakh T.: Ball Structures and Colorings of Graphs and Groups.Mathematical Studies Monograph Series, 11, VNTL Publishers, L'viv, 2003. Zbl 1147.05033, MR 2392704 |
| Reference:
|
[9] Protasov I., Protasova K.: On hyperballeans of bounded geometry.Eur. J. Math. 4 (2018), no. 4, 1515–1520. MR 3866708, 10.1007/s40879-018-0236-y |
| Reference:
|
[10] Protasov I., Protasova K.: The normality of macrocubes and hyperballeans.Eur. J. Math. 7 (2021), no. 3, 1274–1279. MR 4289511, 10.1007/s40879-020-00440-x |
| Reference:
|
[11] Protasov I., Zarichnyi M.: General Asymptology.Mathematical Studies Monograph Series, 12, VNTL Publishers, L'viv, 2007. MR 2406623 |
| Reference:
|
[12] Przesławski K., Yost D.: Continuity properties of selectors in Michael's theorem.Michigan Math. J. 36 (1989), no. 1, 113–134. MR 0989940, 10.1307/mmj/1029003885 |
| Reference:
|
[13] Roe J.: Lectures on Coarse Geometry.University Lecture Series, 31, American Mathematical Society, Providence, 2003. Zbl 1042.53027, MR 2007488, 10.1090/ulect/031/10 |
| . |
