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Abstract Swiss cheese space and classicalisation of Swiss cheeses

Feinstein, Joel; Morley, S.; Yang, H.


S. Morley

H. Yang


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.


Feinstein, J., Morley, S., & Yang, H. (2016). Abstract Swiss cheese space and classicalisation of Swiss cheeses. Journal of Mathematical Analysis and Applications, 438(1), 119-141.

Journal Article Type Article
Acceptance Date Jan 15, 2016
Online Publication Date Feb 6, 2016
Publication Date Jun 1, 2016
Deposit Date Mar 14, 2016
Publicly Available Date Mar 14, 2016
Journal Journal of Mathematical Analysis and Applications
Print ISSN 0022-247X
Electronic ISSN 0022-247X
Publisher Elsevier
Peer Reviewed Peer Reviewed
Volume 438
Issue 1
Pages 119-141
Keywords Swiss Cheeses, Rational Approximation, Uniform Algebras, Bounded Point Derivations, Regularity of R(X)
Public URL
Publisher URL


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