Asymptotic resemblance (Q329919)

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.NEWLINENEWLINETo 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.NEWLINENEWLINEThe 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 \(i=1,2\).NEWLINENEWLINEThe 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.