Inverse limits with irreducible set-valued functions

In this paper, we develop a definition for a class of set-valued functions which will be called irreducible functions. We show that these functions can be used to obtain an indecomposable continuum as an inverse limit, and we give sufficient conditions for two such inverse limits to be homeomorphic. This class of irreducible functions includes all open maps on [ 0 , 1 ] which are not homeomorphisms. The inverse limits of piecewise linear open maps on [ 0 , 1 ] were classified by William Watkins in 1982, and the results of this paper build on those results and expand the class of functions to which they apply.