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An inductive proof of the Feinstein-Heath Swiss cheese ``Classicalisation'' theorem


Author: J. W. D. Mason
Journal: Proc. Amer. Math. Soc. 138 (2010), 4423-4432
MSC (2010): Primary 46J10; Secondary 54H99
DOI: https://doi.org/10.1090/S0002-9939-2010-10447-1
Published electronically: June 16, 2010
MathSciNet review: 2680066
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Abstract: A theory of allocation maps has been developed by J. F. Feinstein and M. J. Heath in order to prove a theorem, using Zorn's lemma, concerning the compact plane sets known as Swiss cheese sets. These sets are important since, as domains, they provide a good source of examples in the theory of uniform algebras and rational approximation. In this paper we take a more direct approach when proving their theorem by using transfinite induction and cardinality. An explicit reference to a theory of allocation maps is no longer required. Instead we find that the repeated application of a single operation developed from the final step of the proof by Feinstein and Heath is enough.


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Additional Information

J. W. D. Mason
Affiliation: School of Mathematical Sciences, University of Nottingham, Nottingham, NG7 2RD, United Kingdom
Email: pmxjwdm@nottingham.ac.uk

DOI: https://doi.org/10.1090/S0002-9939-2010-10447-1
Keywords: Swiss cheeses, rational approximation, uniform algebras
Received by editor(s): October 30, 2009
Received by editor(s) in revised form: February 12, 2010
Published electronically: June 16, 2010
Additional Notes: The author was supported by a Ph.D. grant from the EPSRC (UK)
Communicated by: Nigel J. Kalton
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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