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Generating continuous mappings with Lipschitz mappings


Authors: J. Cichon, J. D. Mitchell and M. Morayne
Journal: Trans. Amer. Math. Soc. 359 (2007), 2059-2074
MSC (2000): Primary 54H15, 20M20
DOI: https://doi.org/10.1090/S0002-9947-06-04026-8
Published electronically: December 15, 2006
MathSciNet review: 2276612
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Abstract: If $ X$ is a metric space, then $ \mathcal{C}_{X}$ and $ \mathcal{L}_X$ denote the semigroups of continuous and Lipschitz mappings, respectively, from $ X$ to itself. The relative rank of $ \mathcal{C}_{X}$ modulo $ \mathcal{L}_{X}$ is the least cardinality of any set $ U\setminus \mathcal{L}_{X}$ where $ U$ generates $ \mathcal{C}_{X}$. For a large class of separable metric spaces $ X$ we prove that the relative rank of $ \mathcal{C}_{X}$ modulo $ \mathcal{L}_X$ is uncountable. When $ X$ is the Baire space $ \mathbb{N}^{\mathbb{N}}$, this rank is $ \aleph_{1}$. A large part of the paper emerged from discussions about the necessity of the assumptions imposed on the class of spaces from the aforementioned results.


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Additional Information

J. Cichon
Affiliation: Institute of Mathematics, Wrocław University of Technology, Wybrzeże Wyspiań- skiego 27, 50-370 Wrocław, Poland
Email: Jacek.Cichon@pwr.wroc.pl

J. D. Mitchell
Affiliation: Mathematics Institute, University of St Andrews, North Haugh, St Andrews, Fife, KY16 9SS, Scotland
Email: jdm3@st-and.ac.uk

M. Morayne
Affiliation: Institute of Mathematics and Computer Science, Wrocław University of Technology, Wybrzeże Wyspiańskiego 27, 50-370 Wrocław, Poland
Email: Michal.Morayne@pwr.wroc.pl

DOI: https://doi.org/10.1090/S0002-9947-06-04026-8
Keywords: Relative ranks, functions spaces, continuous mappings, Lipschitz mappings, Baire space
Received by editor(s): January 28, 2005
Published electronically: December 15, 2006
Article copyright: © Copyright 2006 American Mathematical Society

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