A decomposition theorem for χ1-convex sets

A set S in $\text{R}$ is said to be χ-convex if and only if S does not contain a visually independent subset having cardinality χ. It is natural to ask when an χ-convex set may be expressed as a countable union of convex sets. Here it is proved that if S is a closed χ-convex set in the plane and $\text{R}$ has at most finitely many bounded components, then S is a countable union of convex sets. A parallel result holds in $\text{R}$ when S is a closed χ-convex set which contains all triangular regions whose relative boundaries are in S. However, the result fails for arbitrary χ-convex sets, even in the plane.