On a certain type of homogeneous plane continuum

Abstract

In 1920 very little was known about the class of homogeneous, bounded continua in the plane. At that time Knaster and Kuratowski [1] raised the question:' Is every such (nondegenerate) continuum a simple closed curve? Mazurkiewicz [2] showed such a continuum is a simple closed curve if it is locally connected, and I showed this is the case if the continuum is aposyndetic [3]. H. J. Cohen [4] proved that if a homogeneous, bounded, plane continuum contains a simple closed curve, it is a simple closed curve. And finally I proved that every homogeneous, compact continuum lying in but not separating a plane is indecomposable [5]. So the class of homogeneous, bounded, plane continua may be typed as follows: Type 1. Those which do not separate the plane. (These must all be indecomposable, and continua of Type 1 other than degenerate ones are known to exist [6 and 7].) Type 2. Those which are decomposable. (These must all separate the plane, and continua of Type 2 other than simple closed curves are known to exist [8].) Type 3. Those which separate the plane but are indecomposable. (Whether any of this type exists is not known. However, see [9, Example 2, pp. 48-49].) It is the purpose of this paper to show that each homogeneous, bounded, plane continuum of Type 2 is either a simple closed curve or becomes one under a natural aposyndetic decomposition,2 the elements of the decomposition being mutually homeomorphic continua of Type 1. In other words, thinking of a plane as an upper semicontinuous collection of continua (each lying in but not separating a given plane), every continuum of Type 2 is the sum of the elements of a simple closed curve lying in a plane of elements of Type 1.