On partitions of plane sets into simple closed curves. II

Bankston, Paul

doi:10.1090/S0002-9939-1983-0715874-4

On partitions of plane sets into simple closed curves. II
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Proc. Amer. Math. Soc. 89 (1983), 498-502 Request permission

Addendum: Proc. Amer. Math. Soc. 91 (1984), 658.

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.

References

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Remarque sur un théorème de M. Mullikin

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