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 f i surjective, and upper semicontinuous, and the graph of f i is connected. If n is a positive integer greater than 1, let X n = { 〈 x 0 , … , x n 〉 : x i − 1 ∈ f i ( 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 ∏ i = 0 N I i is connected, the projection of X to ∏ i = N ∞ I i is connected, and the projection of every component of X to I N is I N , then X is connected.