Let S be a compact set in R2. Assume that for every finite set F in bdry S there exist points s and t (depending on F) such that every point of F is via S from at least one of s or /. Then S is a union of two starshaped sets. If clearly visible is replaced by the weaker term visible, then the result fails. 1. Introduction. We begin with some preliminary definitions. Let S be a set in Rd. For points x and y in S, we say x sees y via S (x is from y via S) if and only if the corresponding segment [x, y] lies in S. Point x is from y via S if and only if there is some neighborhood N oΐ x such that y sees each point oί S Γ N via S. Set S is starshaped if and only if there is some point p in S such that p sees each point of S via S, and the set of all such points p is called the (convex) kernel of S. A well-known theorem of KrasnoseΓskii [5] states that if S is a nonempty compact set in Rd9 then S is starshaped if and only if every d + 1 points of S are via S from a common point. Moreover, points of S may be replaced by boundary points of S to produce a stronger result. Other KrasnoseΓskii-type theorems have been obtained for starshaped sets, and in several recent studies ([1], [3], [4]), a helpful tool has been the concept of visible. Here we use the idea of to examine a related problem, that of obtaining a KrasnoseΓskii- type characterization for unions of starshaped sets. Although this kind of problem is mentioned in [8, Prob. 6.6, p. 178] and in [2], it is also closely related to work by Lawrence, Hare, and Kenelly [6] concerning unions of convex sets, and their results will play an important role. Restricting our attention to unions of two starshaped sets in the plane, we establish the following result: Let S be a compact set in R2. Assume that for every finite set F in the boundary of S there exist points s and / (depending on F) such that every point of F is via S from at least one of s or t. Then S is a union of two starshaped sets. If