  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.
