Monotone Decompositions of Hausdorff Continua

Eldon J. Vought
1976 Proceedings of the American Mathematical Society  
A monotone, upper semicontinuous decomposition of a compact, Hausdorff continuum is admissible if the layers (tranches) of the irreducible subcontinua of M are contained in the elements of the decomposition. It is proved that the quotient space of an admissible decomposition is hereditarily arcwise connected and that every continuum M has a unique, minimal admissible decomposition <£. For hereditarily unicoherent continua & is also the unique, minimal decomposition with respect to the property
more » ... ct to the property of having an arcwise connected quotient space. A second monotone, upper semicontinuous decomposition § is constructed for hereditarily unicoherent continua that is the unique minimal decomposition with respect to having a semiaposyndetic quotient space. Then & refines § and S refines the unique, minimal decomposition £ of FitzGerald and Swingle with respect to the property of having a locally connected quotient space (for hereditarily unicoherent continua). For a compact, Hausdorff continuum M, FitzGerald and Swingle [4, p. 37] have obtained a unique monotone, upper semicontinuous decomposition £ whose quotient space (M, £) is semilocally connected and which is minimal with respect to these properties. A second description of this decomposition using collections of closed subsets which separate M is given by them in [4, p. 49] and by McAuley in [9, p. 2]. For a compact, metric continuum M, Charatonik has defined a decomposition to be admissible if it is monotone, upper semicontinuous and the layers of the irreducible subcontinua of M are contained in the elements of the decomposition [2, pp. 115-116]. He then proves in the same paper that the quotient space of an admissible decomposition of a continuum M is hereditarily arcwise connected and that every continuum has a unique, minimal admissible decomposition &. One purpose of this paper is to extend Charatonik's results to compact, Hausdorff continua. A second purpose pertains to the class of hereditarily unicoherent, Hausdorff continua. It is easily seen with simple examples that in general the two decompositions mentioned above are not comparable, i.e., neither refines the other. However, if the continuum M is hereditarily unicoherent then & refines £ (61 g £) since the semilocally connected quotient space (M, £) must then be arcwise connected. In this paper a third decomposition S is constructed which is the unique minimal decomposition with respect to being monotone, upper semicontinuous and having a semiaposyndetic quotient space. The construction involves a collection of closed subsets of M that separate M and satisfies
doi:10.2307/2041640 fatcat:bab3owbkr5fkxpxy5n2dne6stu