Swiss cheese (mathematics)

For other uses, see Swiss cheese (disambiguation).

In mathematics, a Swiss cheese is a compact subset of the complex plane obtained by removing from a closed disc some countable union of open discs, usually with some restriction on the centres and radii of the removed discs. Traditionally the deleted discs should have pairwise disjoint closures which are subsets of the interior of the starting disc, the sum of the radii of the deleted discs should be finite, and the Swiss cheese should have empty interior. This is the type of Swiss cheese originally introduced by the Swiss mathematician Alice Roth.

More generally, a Swiss cheese may be all or part of Euclidean space Rⁿ – or of an even more complicated manifold – with "holes" in it.

References

• Feinstein, J. F.; Morley, S.; Yang, H. (2016). "Abstract Swiss cheese space and classicalisation of Swiss cheeses". Journal of Mathematical Analysis and Applications. 438 (1): 119–141. arXiv:1503.03785. doi:10.1016/j.jmaa.2016.02.004. MR 3462570. S2CID 55614027.
• van den Berg, M.; Bolthausen, E.; den Hollander, F. (2004). "On the volume of the intersection of two Wiener sausages" (PDF). Annals of Mathematics. 159 (2): 741–783. doi:10.4007/annals.2004.159.741.