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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. II


Author: Paul Bankston
Journal: Proc. Amer. Math. Soc. 89 (1983), 498-502
MSC: Primary 54B15; Secondary 57N05
Addendum: Proc. Amer. Math. Soc. 91 (1984), 658.
MathSciNet review: 715874
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Abstract: We answer some questions raised in [1]. In particular, we prove: (i) Let $ F$ be a compact subset of the euclidean plane $ {E^2}$ such that no component of $ F$ separates $ {E^2}$. Then $ {E^2}\backslash F$ can be partitioned into simple closed curves iff $ F$ is nonempty and connected. (ii) Let $ F \subseteq {E^2}$ be any subset which is not dense in $ {E^2}$, and let $ \mathcal{S}$ be a partition of $ {E^2}\backslash F$ into simple closed curves. Then $ \mathcal{S}$ has the cardinality of the continuum. We also discuss an application of (i) above to the existence of flows in the plane.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1983-0715874-4
PII: S 0002-9939(1983)0715874-4
Keywords: Topological partitions, euclidean plane, simple closed curves, continuous flows
Article copyright: © Copyright 1983 American Mathematical Society