Quasi-monotone images of certain classes of continua

Abstract

Let ƒ be a continuous map from a compact metric continuum X onto a continuum Y. Then ƒ is quasi-monotone if, for each subcontinuum K of Y with nonvoid interior, ƒ^-1(K) has a finite number of components and each is mapped onto K by ƒ. Examples of quasi-monotone maps are local homeomorphisms and other finite to one confluent maps. In the following all maps are assumed to be quasi-monotone from X onto Y. A theorem of L. Mohier and J.B. Fugate [1] says that if X is irreducible between two of its points then Y is also irreducible between two of its points. This result is generalized to the following theorem. If X is irreducible about a finite point set A then either Y is irreducible about ƒ(A) or there is a point y in Y such that Y is irreducible about {y}⋃ƒ(A⧹{α}) for each a in A. Another result is that if X is a continuum that is separated by no subcontinuum, i.e., a θ₁-continuum, then Y is a θ₁-continuum or is irreducible between two of its points.