Format of Original
American Mathematical Society
Proceedings of the American Mathematical Society
Original Item ID
doi: 10.1090/S0002-9939-1983-0702301-6; Shelves: QA1 .A5215 Storage S
We investigate the conjecture that the complement in the euclidean plane E2 of a set F of cardinality less than the continuum c can be partitioned into simple closed curves if F has a single point. The case in which F 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 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 E2\F into "rectangular" simple closed curves.