Finitary approximations of coarse structures - Protasov, I. V. (Matematychni Studii, 2021)
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}^′$, we mean any finitary coarse structure $ \mathcal{E}$ such that $ \mathcal{E}\subseteq \mathcal{E}^′$.If $\mathcal{E}^′$ has a countable base and $E[x]$ is finite for each $x\in X$ then $ \mathcal{E}^′$has a cellular finitary approximation $ \mathcal{E}$ such that the relations of linkness on subsets of $( X,\mathcal{E}^′)$ 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.