Approximate resolutions and box-counting dimension

In an earlier paper the authors introduced a new approach using normal sequences and approximate resolutions to study Lipschitz maps between compact metric spaces. In this paper we introduce two kinds of box-counting dimension, which are defined for every compact metric space with a normal sequence and for every approximate resolution of any compact metric space, and investigate their properties. In a special case those notions coincide with the usual box-counting dimension for compact subsets of Rn. Our box-counting dimensions are Lipschitz subinvariant, where Lipschitz maps are in the sense of the earlier paper. Moreover, we obtain fundamental theorems such as the subset theorem, the product theorem and the sum theorem. As an example, for each r with 0⩽r⩽∞, we present a systematic way to construct a compact metric space with an approximate resolution whose box-counting dimension equals r.