Certain homogeneous unicoherent indecomposable continua

Abstract

A simple closed curve is the simplest example of a compact, nondegenerate, homogeneous continuum. If a bounded, nondegenerate, homogeneous plane continuum has any local connectedness property, even of the weakest sort, it is known to be a simple closed curve [1, 2, 3].1 The recent discovery of a bounded, nondegenerate, homogenous plane continuum which does not separate the plane [4, 5] has given substance to the old question as to whether or not such a continuum must be indecomposable. Under certain conditions such a continuum must contain an indecomposable continuum [6]. It is the main purpose of this paper to show that every bounded homogeneous plane continuum which does not separate the plane is indecomposable. NOTATION. If M is a continuum and x is a point of M, U. will be used to denote the set of all points z of M such that M is aposyndetic at z with respect to x.2 It is evident that U. is an open subset of M.