On decomposition of continua into aposyndetic continua

Abstract

Introduction. This paper generalizes certain methods of decomposition of compact metric continua due to R. L. Moore [3; 4] and G. T. Whyburn [9; 10; 13]. While their methods yield acyclic continuous curves, hyperspaces are obtained here which are aposyndetic [1 ] continua. The concept of a con. tinuum being aposyndetic is a generalization of the concept of continuous curves and was introduced in 1941 by F. B. Jones. In compact metric continua, this idea is equivalent to Whyburn's notion of semi-locally-connectedness [14; 2]. A continuum M, i.e., a closed and connected point set, is said to be aposyndetic at a point p with respect to a point x provided that there exists a subcontinuum N of M and an open subset 0 of M such that M-xDNDODp. If M is aposyndetic at a point p with respect to each point x of M-p, then M is said to be aposyndetic at p. It is said that M is aposyndetic if M is aposyndetic at each of its points. In an early paper [10], Whyburn made use of connected cuttings of a compact metric continuum M to obtain a decomposition of M into an acyclic continuous curve. Later, he made use of nonseparated cuttings [13] to obtain a decomposition of a continuous curve into a nondegenerate acyclic continuous curve. Certain of these theorems concerning nonseparated cuttings are generalized in obtaining an aposyndetic decomposition. Moore [3] has obtained decomposition theorems by use of certain sets M(P) defined as follows: For each point P of a compact continuum M, let M(P) denote the set of all points X of M such that there do not exist uncountably many different points each separating P from X in M. He proved the following theorem: If M is a compact metric continuum and G is the collection of all point sets M(P) for all points P of M, then G is an upper semi-continuous collection of disjoint continua filling up M and G is an acyclic continuous curve with respect to its elements as points. The definition of the sets M(P) due to Moore may be generalized in the following manner: Suppose that M is a continuum and that p is a point of M.