On Partitions of Plane Sets Into Simple Closed Curves. II

Abstract

We answer some questions raised in [1]. In particular, we prove: (i) Let $F$ be a compact subset of the euclidean plane ${E^2}$ such that no component of $F$ separates ${E^2}$. Then ${E^2}\backslash F$ can be partitioned into simple closed curves iff $F$ is nonempty and connected. (ii) Let $F \subseteq {E^2}$ be any subset which is not dense in ${E^2}$, and let $\mathcal {S}$ be a partition of ${E^2}\backslash F$ into simple closed curves. Then $\mathcal {S}$ has the cardinality of the continuum. We also discuss an application of (i) above to the existence of flows in the plane.