Concerning the Cut Points of a Continuous Curve when the Arc Curve, AB, Contains Exactly N Independent Arcs

Abstract

In 1925 Menger f introduced the term for those compact connected spaces which have the property that for every point of the set and every e> 0 there exists a neighborhood of diameter less than e which contains the point and has a totally disconnected I boundary. A point of such a set is said to be regular if it is possible to choose these neighborhoods in such a manner that its boundary is finite, while it is of order n if for every e we can find the neighborhood of diameter less than e whose boundary consists of exactly n points. In a subsequent paper he shows that in a regular curve, i. C., a curse containing only regular points, there is corresponding to every point of order n a set of n arcs having the point in common and otherwise distinct.? It should be noted that every regular curve of Menger is also a continuous curve,? that is, it is conilected im kleinen at every poinlt. Of course there are continuous curves which are neither regular nor even curves in the sense defined above. In this paper is colisidered a plane continuous curve in which there are two points a and b which have the property that there are in the continuous curve exactly n arcs frorn, a to b which have no points other than a and b in comrnmon while it is impossible to find a similar set of n + 1 arcs from a to b. Under these hypotheses it is proved that there are exactly n points in the curve whose omission separates a from b in the curve. This theorem provides a sort of converse to the theorem of Menger mentioned. The precise converse of the theorem of this paper is an easy consequence of the theorem itself, and