On Partitions of Plane Sets into Simple Closed Curves

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.