SUM AND DECOMPOSITION THEOREMS FOR CONTINUA IN CLASS WI

In this paper we investigate decomposable atriodic continua and their inverse images. We apply the results of this investigation to prove a sum theorem for continua in class W. We also use these results to prove that atriodic continua which admit a monotone upper semicontinuous decom position to an arc, each element of which is in class W, are in class W. 1. Definitions All spaces considered in this paper are metric. By a continuum we mean a compact connected space. A mapping is a continuous function. If f is a mapping defined on a space X and A c X, denote the restriction of f to A by fiA. Suppose M is a continuum, H is a subcontinuum of M and f is a mapping of a continuum onto M. Following Ingram [5] we say that f is confluent with respect to H (respectively, weakly confluent with respect to H) provided that for each (resp., some) component, L, of f-l(H), f(L) = H. We say that f is confluent (resp., weakly confluent) provided f is confluent (resp., weakly confluent) with respect to each subcontinuum of M. The continuum M is in class W provided lWith the exception of Theorem 6, the results in this paper are from the author's doctoral dissertation, written under the direction of W. T. Ingram at the University of Houston. 242 Davis that for each mapping f of a continuum onto M, f is weakly confluent. A continuum is a triod provided it contains a subcon tinuum whose complement has at least three components. A continuum is atriodic if it contains no triode The con tinuum M is unicoherent provided that if A and Bare sub-continua of M such that M = A U B then A n B is connected. 2. Decomposable Atriodic Continua Throughout sections 2 and 3 we adopt the following standing hypothesis: Suppose that Hand K are continua which intersect, X = H U K, X is atriodic, H n K is con nected, and that each of Hand K contains a point which is not in the other