Concerning Connected and Regular Point Sets

In this paper an extension will be given of a theorem of the author'sf which states that if A and B are any two points of a continuous curve M, and K denotes the set of all those points of M which separate} A and B in M, then K+A+B is a closed set of points. I t will be shown that if A and B are any two points of any connected and regular (connected im kleinen) point set M, and K denotes the set of all those points of M which separate A and B in M, then K+A+B is a closed and bounded set of points. This extended theorem is applied to show that a simple continuous arc may be defined as a connected and regular point set which is irreducibly connected! between some two of its points.