On the existence of acyclic curves satisfying certain conditions with respect to a given continuous curve

Part I of this paper has to do with connected sets of cut points of a given continuous curve. It will be shown that any connected set of cut points, K, of a given continuous curve, M, lie together in an arc of K which is a subset of the boundaries of a finite number of complementary domains of M. G. T. Whyburn calls attention to the fact that from his results it follows that K is arc-wise connected. To show that any two points of K can be joined by an arc of K which is a subset of the boundaries of a finite number of complementary domains of M is the object of Part I. Part II has to do with a totally disconnected closed subset, K, of a given continuous curve, M, no subset of which disconnects M. R. L. Moore has shown that any two points not belonging to a bounded continuous curve can be joined by an arc which does not disconnect the continuous curve. The object of Part II of this paper is to show that if M is a bounded continuous curve which contains no domain and K is a totally disconnected closed subset of M, no subset of which disconnects M, then there exists an acyclic continuous curve, T, containing K and having in common with M a totally disconnected set, the omission of which leaves M connected, and such that the end points of T is a subset of KMathematic