Connectedness and Ingram-Mahavier products

We introduce a new tool which we call an Ingram-Mahavier product to aid in the study of inverse limits with set-valued functions, and with this tool, obtain some new results about the connectedness properties of these inverse limits. Suppose X=lim⟵(Ii,fi) is an inverse limit with set-valued functions fi over intervals Ii with each fi surjective, and upper semicontinuous, and the graph of fi is connected. If n is a positive integer greater than 1, let Xn={〈x0,…,xn〉:xi−1∈fi(xi),i>0}. We show, with the help of the Mountain Climbing Theorem, that (1) Xn is never totally disconnected, and (2) if for some factor N, which is not the first factor, the projection of X to ∏i=0NIi is connected, the projection of X to ∏i=N∞Ii is connected, and the projection of every component of X to IN is IN, then X is connected.