Monotone decompositions of continua not separated by any subcontinua

Abstract

Let M be a compact, metric continuum that is separated by no subcontinuum. If such a continuum has a monotone, upper semicontinuous decomposition, the elements of which have void interior and for which the quotient space is a simple closed curve, then it is said to be of type A ′ {\text {A}}’ . It is proved that a bounded plane continuum is of type A’ {\text {A’}} if and only if M contains no indecomposable subcontinuum with nonvoid interior. In E 3 {E^3} this condition is not sufficient and an example is given to illustrate this. However, it is shown that if M is hereditarily decomposable then M is of type A ′ {\text {A}}’ . Next, a condition is given that characterizes continua of type A’ {\text {A’}} . Also the structure of the elements in the decomposition of a continuum of type A’ {\text {A’}} is discussed and the decomposition is shown to be unique. Finally, some consequences of these results and some remarks are given.