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

TL;DR

This paper provides an inductive proof of the Feinstein-Heath Swiss cheese classicalisation theorem, simplifying previous methods by eliminating the need for allocation maps and using transfinite induction and cardinality.

Contribution

It introduces a more direct inductive proof of the theorem, removing the reliance on allocation maps and employing a single operation with transfinite induction.

Findings

Simplified proof of the classicalisation theorem

Elimination of allocation maps in the proof process

Use of transfinite induction and cardinality techniques

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.