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(I-i, f(i)) is an inverse limit with set-valued functions f(i) over intervals I-i, with each L surjective, and upper semicontinuous, and the graph of fi is connected. If n is a positive integer greater than 1, let X = {(x(0), " " " E fi(x(i)), i > 0}. We show, with the help of the Mountain Climbing Theorem, that (1) X-n, is never totally disconnected, and (2) if for some factor N, which is not the first factor, the projection of X to Pi(N)(i=0) h is connected, the projection of X to Pi(infinity)(i=N) I is connected, and the projection of every component of X to I-N is IN, then X is connected. (C) 2014 Elsevier B.V. All rights reserved.