A Planar 3-Convex Set is Indeed a Union of Six Convex Sets

Suppose S is a planar set. Two points $a,b$ in S  see each other via S if $a,b]$ is included in S . F. Valentine proved in 1957 that if S is closed, and if for every three points of S, at least two see each other via S, then S is a union of three convex sets. The pentagonal star shows that the number three is the best possible. We drop the condition that S is closed and show that S is a union of (at most) six convex sets. The number six is best possible.