## Asymptotic resemblance.(English)Zbl 1355.53019

Proximity relations as studied by Efremovic allow one to model the notion of nearness for subsets of a set, $$X$$. Often $$X$$ is a metric space, and the relation is thought of as revealing the ‘small scale’ structure of $$X$$. This has a variety of uses.
To study the ‘large scale’ structure of spaces, for instance in coarse geometry, one often uses the notion of a coarse structure on $$X$$, as defined by Roe.
The aim of this paper is to introduce and study a large scale counterpart of proximity, which is given the name of ‘asymptotic resemblance’. If $$X$$ is a set, a binary relation $$\lambda$$ on the powerset of $$X$$ is called an asymptotic resemblance if it is an equivalence relation on the powerset of $$X$$ and, in addition, it satisfies two conditions: (i) if $$A_1\lambda B_1$$ and $$A_2\lambda B_2$$ then $$(A_1\cup A_2)\lambda (B_1\cup B_2)$$, and (ii) if $$(B_1\cup B_2)\lambda A$$ and $$B_1$$, $$B_2$$ are non empty, then there are non-empty subsets $$A_1$$ and $$A_2$$ such that $$A=A_1\cup A_2$$ and $$B_i\lambda A_i$$ for $$i=1,2$$.
The properties of this resemblance relation are explored in some depth. Several generic families of examples are given and the relationship with coarse structures is examined.

### MSC:

 53C05 Connections (general theory) 18B30 Categories of topological spaces and continuous mappings (MSC2010) 51F99 Metric geometry 54C20 Extension of maps
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### References:

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