# Relative ranks of Lipschitz mappings on countable discrete metric spaces

J. Cichon, James D. Mitchell, Michal Morayne, Yann Peresse

Research output: Contribution to journalArticlepeer-review

2 Citations (Scopus)

## Abstract

Let X be a countable discrete metric space and let XX 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 XX such that every element of XX 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.
We show 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 NN. 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 Nk and the sequence of partial sums of the harmonic series with the usual euclidean metric.
To conclude, 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.
Original language English 412-423 12 Topology and its Applications 158 3 3 Dec 2010 https://doi.org/10.1016/j.topol.2010.11.020 Published - 15 Feb 2011

## Keywords

• relative rank
• function space
• continuous mapping
• Liptschitz mapping
• Semigroups
• discrete space

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