Abstract
Swiss cheese sets are compact subsets of the complex plane obtained by deleting a sequence of open disks from a closed disk. Such sets have provided numerous counterexamples in the theory of uniform algebras. In this paper, we introduce a topological space whose elements are what we call “abstract Swiss cheeses”. Working within this topological space, we show how to prove the existence of “classical” Swiss cheese sets (as discussed in [6]) with various desired properties. We first give a new proof of the Feinstein–Heath classicalisation theorem [6]. We then consider when it is possible to “classicalise” a Swiss cheese while leaving disks which lie outside a given region unchanged. We also consider sets obtained by deleting a sequence of open disks from a closed annulus, and we obtain an analogue of the Feinstein–Heath theorem for these sets. We then discuss regularity for certain uniform algebras. We conclude with an application of these techniques to obtain a classical Swiss cheese set which has the same properties as a non-classical example of O'Farrell [5].
Highlights
Throughout, we use the term compact plane set to mean a non-empty, compact subset of the complex plane
A Swiss cheese set is a compact subset of C obtained by deleting a sequence of open disks from a closed disk
Such sets have been used as examples in the theory of uniform algebras and rational approximation
Summary
Throughout, we use the term compact plane set to mean a non-empty, compact subset of the complex plane. We give an example of the application of a combination of these results to construct an example of a classical Swiss cheese set X such that R(X) is regular and admits a non-degenerate bounded point derivation of infinite order (as defined in Section 8), which improves an example of O’Farrell [5]. This fits into our general classicalisation scheme
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