Concerning Simple Continuous Curves and Related Point Sets

Abstract

In his paper Concerning Simple Continuous Curves, * R. L. Moore applied the term simple continouou curve to any point set which is either an arc, a simple closed curve, an open curve or a ray, and gave various topological characterizations for a point set that falls within the general class of such curves, as well as for the individual types of curves.f The intent of the present paper is to supplement the work of Moore just cited, extending two of his results to more general spaces, and giving certain new definitions which the author believes may be of value.$ As Moore made clear in his paper, the requirement of boundedness in a definition of arc may introduce great difficulties in certain problems. It will be noted that none of the definitions of arc, simple closed curve, etc., given in the present instance makes use of this condition. Furthermore, the condition that the set in question be closed is eliminated in several cases-a feature that would seem to be of importance in problems concerning non-closed sets. To summarize briefly, before proceeding to the demonstrations: In ? 1, Moore's characterization of simple continuous curves as a class is extended to any euclidean space, and is then applied to establish a new characterization based on a certain type of set which the author has found useful in other connections. In ? 2 definitions are given of arc, the more general class of sets known as irredtcible connexes,? and of simple closed and quasi-closed curves. In this connection it is indicated how a simple proof may be obtained of the theorem ? that every interval of an open curve is an arc; also a