On partitions of plane sets into simple closed curves

Abstract

We investigate the conjecture that the complement in the euclidean plane E 2 {E^2} of a set F F of cardinality less than the continuum c c can be partitioned into simple closed curves iff F F has a single point. The case in which F 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 F either has finitely many isolated points or finitely many cluster points. Also we show there exists a self-dense totally disconnected set F F of cardinality c c and a partition of E 2 ∖ F {E^2}\backslash F into "rectangular" simple closed curves.