Finitary approximations of coarse structures
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 $...
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| Format: | Article |
| Language: | deu |
| Published: |
Ivan Franko National University of Lviv
2021-03-01
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| Series: | Математичні Студії |
| Subjects: | |
| Online Access: | http://matstud.org.ua/ojs/index.php/matstud/article/view/127 |
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| Summary: | 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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| ISSN: | 1027-4634 2411-0620 |