(L,⁎)-valued coarse structures on powersets

Abstract

The main purpose of this paper is to establish a fundamental framework of L-valued coarse structures. At first an approach to fuzzification of the notion of coarse structure is proposed. An L-valued coarse structure on a set X as an L-fuzzy subset U of the powerset 2X×X satisfying L-valued analogues of the axioms of a coarse structure is given. Some basic properties of L-valued coarse structure spaces are studied. Then we introduce the notion of asymptotic dimension in [0,1]-valued coarse spaces and show that the degree of asymptotic dimension is invariant under fuzzy coarse equivalences.