Projects per year
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 Sierpinski 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 vertical bar A vertical bar <= 1. On the other hand, we give several results relating vertical bar A vertical bar to the cardinal 0; defined as the minimum cardinality of a dominating family for NN. In particular, we give a condition on the metric of X under which vertical bar A vertical bar >= partial derivative holds and a further condition that implies vertical bar A vertical bar <= partial derivative. 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 vertical bar A vertical bar = 1 or almost always, in the sense of Baire category, vertical bar A vertical bar = partial derivative. (C) 2010 Elsevier B.V. All rights reserved.
Original language  English 

Pages (fromto)  412423 
Number of pages  12 
Journal  Topology and Its Applications 
Volume  158 
Issue number  3 
Early online date  3 Dec 2010 
DOIs  
Publication status  Published  15 Feb 2011 
Keywords
 Relative rank
 Function space
 Continuous mapping
 Lipschitz mapping
 Semigroups
 Discrete space
 SEMIGROUPS
 SETS
Fingerprint
Dive into the research topics of 'Relative ranks of Lipschitz mappings on countable discrete metric spaces'. Together they form a unique fingerprint.Projects
 1 Finished

Semigroups of Mappings OTG EPG0169921: Semigroups of Mappings: Set Theoretic Analytic and Combinatorial Aspects
1/09/08 → 31/01/10
Project: Standard