On partitions of plane sets into simple closed curves

Bankston, Paul

doi:10.1090/S0002-9939-1983-0702301-6

On partitions of plane sets into simple closed curves
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Proc. Amer. Math. Soc. 88 (1983), 691-697 Request permission

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.

References

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Remarque sur un théorème de M. Mullikin

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Anna M. Mullikin, Certain theorems relating to plane connected point sets, Trans. Amer. Math. Soc. 24 (1922), no. 2, 144–162. MR 1501219, DOI 10.1090/S0002-9947-1922-1501219-4
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