Suppose that K is the sum of two circles in the plane tangent at the x. (They may be internally or externally tangent.) Let D be the complementary domain of K having K as its boundary; let Q be one of the other complementary domains; and let M be the complement of D +Q. If p is a of M not in Q and q is a of Q, it is clear that x cuts p from q in M+Q. In fact, since M is locally connected, the existence of such a follows from a wellknown theorem in plane topology.2 However, if, instead of being a locally connected continuum, K is merely a continuum (possibly indecomposable), then M is not necessarily locally connected and the existence of such a is not so evident or its ambiguous location gives no clue to the proof of its existence. The purpose of this paper is to state and prove a existence theorem of this general nature. As is frequently the case with theorems, when one weakens the hypothesis by discarding the local connectedness requirement, one must weaken the conclusion by replacing the notion of separating point by the notion of cut But since the two notions are equivalent in the presence of local connectedness, one usually gets a stronger theorem, and this is the case here. DEFINITIONS AND NOTATION. Let space be a 2-sphere and let S denote the set of all points of the space.3 If a and b are points of a continuum M, the subset X of M-(a+b) is said to a from b in M provided that every subcontinuum of M containing a and b contains a of X. When X contains only one point, it is called a point. If M is a continuum, by a complementary domain of M is meant a component of S-M. If D is a connected open set and Q is a connected subset of S-D, by the outer boundary of D with respect to Q is meant the boundary of the complementary domain of D containing Q.