Monotone Decompositions of Iuc Continua

Abstract

Abstract. For the class of hereditarily unicoherent metric continua a spectrum of monotone decompositions has been developed by several authors which " improves" the quotient spaces. This spectrum is developed for a broader class of continua, namely continua with property IUC. A metric continuum M has property IUC provided each proper subcontinuum of M with interior is unicoherent. One important result which develops is that semiaposyndetic IUC continua are hereditarily arcwise connected. Also the notion of smoothness is studied for IUC continua.0. Introduction. In [5] FitzGerald and Swingle, by use of set functions, described a monotone upper semicontinuous decomposition 6Da of a compact Hausdorff continuum M such that sHa is the core decomposition of M with respect to having an aposyndetic quotient space. A spectrum of decompositions which " fills in" between M and M/6i)a has been developed for hereditarily unicoherent metric continua by such authors as J. J. Charatonik [1], E. J. Vought [8, 12] and G. R. Gordh [6,8].The main purpose of this paper is to extend the spectrum of decompositions to the class of IUC continua. A compact metric continuum has property IUC provided each proper subcontinuum with interior is unicoherent [3]. If the continuum has property IUC hereditarily then the continuum has property HIUC. Note that the class of IUC continua includes all atriodic continua and hereditarily unicoherent continua.Throughout this paper M will denote a compact metric continuum. An upper semicontinuous (u.s.c.) decomposition ty of M is core with respect to some property P provided tf) is the unique minimal decomposition with respect to which ty has property P. A set function TV is expansive provided that if A and B are subsets of M then A E N(A) and N(A) E N(B) whenever A E B. The subset A is TV-closed provided A = N(A). In [5, Theorem 2.5, p. 37] FitzGerald and Swingle prove that if TV is any expansive set function on M then there exists a core decomposition § of M with respect to the property: § is u.s.c with TV-closed elements. They also note that if TV is monotone then § is monotone.Let <î> be a u.s.c. decomposition of M and % E 6Í). We denote by %* the subset of M consisting of the sum of the elements of %. If B is a subset of M/ty then B~x will