Finitary approximations of coarse structures

  • I. V. Protasov Taras Shevchenko National University of Kyiv, Kyiv, Ukraine
Keywords: finitary coarse structure, finitary approximation, linkness

Abstract

A coarse structure $ \mathcal{E}$ on a set $X$ is called finitary if, for each entourage $E\in \mathcal{E}$, there exists a natural number $n$ such that $ E[x]< n $ for each $x\in X$. By a finitary approximation of a coarse structure $ \mathcal{E}^\prime$, we mean any finitary coarse structure $ \mathcal{E}$ such that $ \mathcal{E}\subseteq \mathcal{E}^\prime$.
If $\mathcal{E}^\prime$ has a countable base and $E[x]$ is finite for each $x\in X$ then $ \mathcal{E}^\prime$
has a cellular finitary approximation $ \mathcal{E}$ such that the relations of linkness on subsets of $( X,\mathcal{E}^\prime)$ and $( X, \mathcal{E})$ coincide.
This answers Question 6 from [8]: the class of cellular coarse spaces is not stable under linkness. We define and apply the strongest finitary approximation of a coarse structure.

References

T. Banakh, Small uncountable cardinals in large-scale topology, preprint, arXiv: 2002.08800.

T. Banakh, I. Banakh, On the dimension of coarse spaces, preprint, arXiv: 2001.04300.

T. Banakh, I. Protasov, Set-theoretical problems in Asymptology, preprint, arXiv: 2004.01979.

I. V Protasov, Balleans of bounded geometry and G-spaces, Algebra Discrete Math., 7 (2008), №2, 101–108.

I. Protasov, Varieties of coarse spaces, Axioms, 7 (2018), №2, 32. https://doi.org/10.3390/axioms7020032.

I. Protasov, Decompositions of set-valued mappings, preprint, arXiv: 1908.03911.

I. Protasov, T. Banakh, Ball structures and colorings of groups and graphs, Math. Stud. Monogr. Ser., 11, VNTL, Lviv, 2003.

I. Protasov, K. Protasova, Closeness and linkness in balleans, Mat. Stud. 53 (2020), №1, 100–108. https://doi.org/10.30970/ms.53.1.100-108

I. Protasov, M. Zarichnyi, General Asymptology, Math. Stud. Monogr. Ser., V.12, VNTL, Lviv, 2007.

J. Roe, Lectures on coarse geometry, Univ. Lecture Ser., 31, American Mathematical Society, Providence RI, 2003.

Published
2021-03-04
How to Cite
1.
Protasov IV. Finitary approximations of coarse structures. Mat. Stud. [Internet]. 2021Mar.4 [cited 2021Apr.15];55(1):33-6. Available from: http://matstud.org.ua/ojs/index.php/matstud/article/view/127
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Articles