Finitary approximations of coarse structures

I. V.  Protasov

doi:10.30970/ms.55.1.33-36

I. V. Protasov Taras Shevchenko National University of Kyiv, Kyiv, Ukraine

DOI: https://doi.org/10.30970/ms.55.1.33-36

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.

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