On non-cut sets of locally connected continua
Wilfred M. Kincaid · Bulletin of the American Mathematical Society · 1943
have proved independently that if a locally connected continuum S contains a non-cut point p, there exists an arbitrarily small region R containing p and such that S -R is connected.Our paper is concerned with certain generalizations of this theorem.We shall consider a space 5 which is a locally connected continuum and contains a closed set P such that S -P is connected.We show that under these hypotheses P can be enclosed in an open set R, the sum of a finite number of regions, whose complement is a locally connected continuum.We show further that if there exists a family of sets % no element of which separates S -P, then there exist two open sets R and R' (with RZ}R'Z)P) of the above type and having the property that no element of $ contained in S -R separates S -R'.When the elements of $ are single points, it is possible to choose R' = R; but this is not possible in the more general case.We close by showing that if 5 is not separated by any element of % plus any set of n points, and if Q is the sum of n sets of sufficiently small diameter and having sufficiently great mutual distances, then the set S -Q has at most one component whose diameter is greater than a preassigned positive quantity, and this component is not separated by any element of % at a sufficiently great distance from Q.We recall some well known results.3 Let M be a locally connected continuum.Then :(1) M is a metric space having property S. A (2) M is the sum of a finite number of arbitrarily small connected Presented to the Society September 10, 1942; received by the editors July 31,1942. 1 See W. L. Ayres, On continua which are disconnected by the omission of a point and some related problems, Monatshefte für Mathematik und Physik vol.36 (1929) pp.135-147.The theorem quoted here corresponds to Theorem 2 p. 149.2 See H. M. Gehman, Concerning certain types of non-cut points, with an application to continuous curves, Proc.