On Partitions of Plane Sets into Simple Closed Curves

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 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 \F into "rectangular" simple closed curves. Let X and Y be topological spaces. By a topological partition of Y into copies of X we mean a covering of Y by pairwise disjoint sets, each of which is homeomorphic with X. In this note Y will be a subset of the euclidean plane E2 and X will be the unit circle. A homeomorphic copy of X will be referred to, as usual, as a simple closed curve (sec). In [1] it is proved that, among the compact connected two-manifolds, only the torus and the Klein bottle can be partitioned into scc's. The key to the proof is the following 1. Theorem (lemma in proof of Theorem 3.3 in [1]). Let FEE2 be finite. Then E2\F can be partitioned into scc's iff F has exactly one point. Here we are interested in extending Theorem 1; in particular we believe the following is true. 2. Conjecture. Let FEE2 be an infinite set of cardinality less than the continuum c. Then £2\F cannot be partitioned into scc's. In support of this conjecture we have the following result. 3. Theorem. Let F E E2 be an infinite set of cardinality less then c. If either: (i) F has finitely many isolated points; or (ii) F has finitely many cluster points in E2, then E \F cannot be partitioned into sec 's. In order to prove (3), we will need some auxiliary machinery. If S E E2 is any sec, let B(S) denote the bounded complementary domain of S. By the Jordan Curve Theorem, 5 is the common boundary of both B(S) and Received by the editors August 23, 1982 and, in revised form, October 28, 1982. 1980 Mathematics Subject Classification. Primary 54B15, 57N05.