Inverse limits with irreducible set-valued functions.

We define a class of set-valued functions called irreducible functions and show that their inverse limits are indecomposable continua. We go on to further explore this class of inverse limit spaces. This includes a characterization of chainability and a characterization of endpoints of inverse limits of certain irreducible functions. Additionally, we develop multiple tools for determining when two inverse limits of irreducible functions are or are not homeomorphic. This culminates in a homeomorphic classification of the inverse limits of four specific families of irreducible functions.