Valentine (1, Theorems 2 and 3) has defined a three-point property which he called P3 and has shown that a closed subset of the euclidean plane possessing this property is expressible as the union of at most three convex sets. He also showed that if the number of isolated points of local non-convexity of such a set is one, finite and even, or infinite, the set is the union of two convex sets. In this paper we give properties which, together with Valentine's results, characterize those subsets of a plane which may be represented as a union of two closed, convex sets.