Property a and asymptotic dimension

Abstract

The purpose of this note is to characterize the asymptotic dimension asdim(X) of metric spaces X in terms similar to Property A of Guoliang Yu. We prove that for a metric space (X, d) and n ≥ 0 the following conditions are equivalent: a. asdim(X, d) ≤ n. b. For each R, > 0 there is S > 0 and finite non-empty subsets Ax ⊂ B(x, S) × N,x ε X, such that |AxδAy| |Ax∩Ay| < if d(x, y) < R and the projection of Ax onto X contains at most n + 1 elements for allx ε X. c. For each R > 0 there is S > 0 and finite non-empty subsets Ax ⊂ B(x, S) × N,x ε X, such that |AxδAy| |Ax∩Ay| < ε if d(x, y) < R and the projection of Ax onto X contains at most n + 1 elements for allx ε X.