Monotone decompositions into trees of Hausdorff continua irreducible about a finite subset

Abstract

This paper deals with characterizing two types of monotone, upper semi-continuous decompositions of a Hausdorff continuum that is irreducible about a finite subset One of the decompositions is minimal with respect to the property of having a quotient space which is a tree (a hereditarily unicoherent, locally connected continuum) and is characterized in terms of certain collections of subcontinua. The other decomposition is not only minimal but also unique with respect to the properties that the quotient space is a tree and the elements of the decomposition have void interiors. This decomposition is characterized quite simply by prohibiting the existence of indecomposable subcontinua with nonvoid interiors. The structure of the elements of the decompositions that have void interiors is very nice and is described by means of the aposyndetic set function Γ In the case where elements exist with nonvoid interiors, the structure can be very complicated and a final result deals with this structure under some rather stringent conditions.