The converse of the theorem concerning the division of a plane by an open curve

Abstract

In his paper, On the foutndations of plane analysis situs,t R. L. Moore proved that if 1 is an open curvet and S is the set of all points, then S 1 = Si + S2, where Si and S2 are connected point sets such that every arc from a point of Si to a point of S2 contains at least one point of 1.? Clearly the sets Si and S2 are non-compact.11 Professor Moore's theorem is proved on the basis of his set of axioms 13. Thus the theorem is true in certain spaces which are neither metrical, descriptive, nor separable. This theorem for open curves is analogous to the theorem of Jordan,? that a simple closed curve lying wholly within a plane decomposes the plane into an inside and an outside region. The converse of this theorem for simple closed curves was first formulated by Schoenflies,** who makes use of metrical properties in his proof. A different proof has been given by Lennes,tt who uses straight lines. R. L. Moore has pointed out that, on the basis of 13, an argument similar in large part to that of Lennes can be carried through with the use of arcs and closed curves. The object of the present paper is to show that the converse of the open curve theorem holds in spaces satisfying 13. The statement of the converse theorem is as follows: