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

Published electronically:
June 16, 2010

MathSciNet review:
2680066

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Abstract | References | Similar Articles | Additional Information

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.