Finitary approximations of coarse structures. (English) Zbl 1475.22001
Let \((X,\mathcal E)\) be a coarse space. A subset \(\mathcal E'\subseteq \mathcal E\) is called a base of \((X,\mathcal E)\) if for every \(E\in\mathcal E\) there exists \(E'\in \mathcal E'\) such that \(E\subseteq E'\). A coarse space \((X,\mathcal E)\) is called cellular if it has a base consisting of equivalences. (More precise definitions are given in the paper under review and in [the author and K. Protasova, Mat. Stud. 53, No. 1, 100–108 (2020; Zbl 1437.54026)]).
In the paper the notion of a \(\lambda\)-stable class of coarse spaces is studied.
In [loc. cit.] the following question was raised: is the class of cellular spaces \(\lambda\)-stable?
In the given note this question is answered in the negative in ZFC. The authors mention that the question has been solved under some set-theoretical assumptions by the author and K. Protasova [loc. cit.].
In the paper the notion of a \(\lambda\)-stable class of coarse spaces is studied.
In [loc. cit.] the following question was raised: is the class of cellular spaces \(\lambda\)-stable?
In the given note this question is answered in the negative in ZFC. The authors mention that the question has been solved under some set-theoretical assumptions by the author and K. Protasova [loc. cit.].
Reviewer: Mihail I. Ursul (Oradea)
MSC:
| 22A05 | Structure of general topological groups |
| 54E05 | Proximity structures and generalizations |
| 54E35 | Metric spaces, metrizability |
Citations:
Zbl 1437.54026References:
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