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

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.
Comment: 10 pages, 1 figure, Accepted for publication by the Proceedings of the American Mathematical Society, volume 138, year 2010, number 12, pages 4423-4432