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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

On partitions of plane sets into simple closed curves


Author: Paul Bankston
Journal: Proc. Amer. Math. Soc. 88 (1983), 691-697
MSC: Primary 54B15; Secondary 57N05
MathSciNet review: 702301
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Abstract: We investigate the conjecture that the complement in the euclidean plane $ {E^2}$ of a set $ F$ of cardinality less than the continuum $ c$ can be partitioned into simple closed curves iff $ F$ has a single point. The case in which $ F$ is finite was settled in [1] where it was used to prove that, among the compact connected two-manifolds, only the torus and the Klein bottle can be so partitioned. Here we prove the conjecture in the case where $ F$ either has finitely many isolated points or finitely many cluster points. Also we show there exists a self-dense totally disconnected set $ F$ of cardinality $ c$ and a partition of $ {E^2}\backslash F$ into "rectangular" simple closed curves.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1983-0702301-6
PII: S 0002-9939(1983)0702301-6
Keywords: Topological partitions, euclidean plane, simple closed curves
Article copyright: © Copyright 1983 American Mathematical Society