An inductive proof of the Feinstein-Heath Swiss cheese “Classicalisation” theorem

Mason, J.

doi:10.1090/S0002-9939-2010-10447-1

An inductive proof of the Feinstein-Heath Swiss cheese “Classicalisation” theorem
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by J. W. D. Mason PDF

Proc. Amer. Math. Soc. 138 (2010), 4423-4432 Request permission

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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