Concerning simple continuous curves

Abstract

Various definitions of simple continuous arcs and closed curves have been given.t The definitions of arcs usually contain the requirement that the point-set in question should be bounded. In attempting to prove that every interval t of an open curve as defined in a recent papert is a simple continuous arc, while I found it easy to prove that t satisfies all the other requirements of Janiszewski's definition (modified as indicated below) it was only by a rather lengthy and complicated argument that I succeeded in proving that it satisfies the requirement of boundedness. In Lennes' definition the requirement of boundedness is superfluous.? However I found it difficult to prove that t satisfies a certain one of the other requirements of this definition, namely that the point-set in question should contain no proper connected subset that contains both A and B. In the present paper I will give a definition l of a simple continuous arc which stipulates neither that the set 111 should be bounded nor that it should contain no proper connected subset containing both A and B. I will show that, in a euclidean space of two dimensions, every point-set that satisfies this definition is an arc in the sense of Jordan. It is easy to prove? that every interval of an open curve satisfies this definition.