On partitions of plane sets into simple closed curves
Proc. Amer. Math. Soc. 88 (1983), 691-697
Primary 54B15; Secondary 57N05
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Abstract: We investigate the conjecture that the complement in the euclidean plane of a set of cardinality less than the continuum can be partitioned into simple closed curves iff has a single point. The case in which is finite was settled in  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 either has finitely many isolated points or finitely many cluster points. Also we show there exists a self-dense totally disconnected set of cardinality and a partition of into "rectangular" simple closed curves.
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