A theorem on the accessibility of boundary parts of an open point set

Abstract

The theorem was established and proved in order to bridge a gap in the proof of a classical theorem on surfaces z(x, y) of nonpositive Gaussian curvature. Theorem 1 might, however, be of interest in other respects. The completed proof of the differential geometric theorem is given in the subsequent paper.-The special case of Theorem 1 needed for the completion is that where the boundaries of the open set Q and of the open and connected sets Q'CQ have only one or two points in common. None of these points need, of course, be accessible from Q'. Theorem 1, however, ascertains that each of these points is accessible from Q. The conclusion of the theorem remains valid if the open set Q'CQ, instead of being connected, is merely supposed to be connected to the vicinity of C. We call an open set Q' connected to the vicinity of a closed set S if there is a neighborhood N* of S (open set containing S) with the following property. For every neighborhood N of S there is a Jordan arc whose interior lies in Q' and whose end points lie in N and on the boundary of N*, respectively. The generalized Theorem 2 is found to be more easily proved than Theorem 1. It is convenient to extend the notion of accessibility of a closed set S from an open set Q to arbitrary closed sets (not necessarily parts of the boundary of Q).