Closed generalized Mazurkiewicz sets are curves

Mazurkiewicz proved the existence of a subset of the Euclidean plane E2 with the property that every straight line intersects it in exactly two points. A set with this property is called a Mazurkiewicz set. A nondegenerate subset X of E2 is a generalized Mazurkiewicz set if each line that separates two points of X intersects X in exactly two points. We prove that a generalized Mazurkiewicz set must be a simple closed curve if it contains an arc. From this we deduce that a closed, generalized Mazurkiewicz set is a simple closed curve. Simple closed curves in E2 are generalized Mazurkiewicz sets if and only if they bound convex disks.