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 ${\text {A}}’$. It is proved that a bounded plane continuum is of type ${\text {A’}}$ if and only if M contains no indecomposable subcontinuum with nonvoid interior. In ${E^3}$ 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 ${\text {A}}’$. Next, a condition is given that characterizes continua of type ${\text {A’}}$. Also the structure of the elements in the decomposition of a continuum of type ${\text {A’}}$ is discussed and the decomposition is shown to be unique. Finally, some consequences of these results and some remarks are given.