Relative ranks of Lipschitz mappings on countable discrete metric spaces

Let X be a countable discrete metric space and let X X denote the family of all functions on X . In this article, we consider the problem of finding the least cardinality of a subset A of X X such that every element of X X is a finite composition of elements of A and Lipschitz functions on X . It follows from a classical theorem of Sierpiński that such an A either has size at most 2 or is uncountable. w that if X contains a Cauchy sequence or a sufficiently separated, in some sense, subspace, then | A | ≤ 1 . On the other hand, we give several results relating | A | to the cardinal d ; defined as the minimum cardinality of a dominating family for N N . In particular, we give a condition on the metric of X under which | A | ≥ d holds and a further condition that implies | A | ≤ d . Examples satisfying both of these conditions include all subsets of N k and the sequence of partial sums of the harmonic series with the usual euclidean metric. clude, we show that if X is any countable discrete subset of the real numbers R with the usual euclidean metric, then | A | = 1 or almost always, in the sense of Baire category, | A | = d .