Generic inverse limits with set-valued functions

Suppose that for each i >= 0, Iiis an interval, and for each i >= 1, Piis a pseudoarc contained in Ii-1 x I-i such that pi P-i-1(i)= Ii-1 and pi P-i(i)= I-i(pi(i-1)and pi(i)denote the respective projections of P-i to the intervals Ii-1 and I-i). Then for each i >= 1, there is a surjective upper semicontinuous map f(i) : Ii -> 2(Ii-1)such that the graph Gamma(f(i)) = P-i. We prove that the inverse limit space (lim) under left arrow (I-i, f(i)): ={(x(0), x(1),...) is an element of Pi (infinity)(i=0) I-i : for each i >= 1, x(i-1) is an element of f(i)(x(i))} is hereditarily disconnected (i.e., no closed nondegenerate sub-generalized inverse limit is connected). It does contain, however, nondegenerate continua. Furthermore, in the space of all such inverse limits in the Hilbert cube, those inverse limits that are formed with pseudoarc bonding maps (we call them pseudoarc generalized inverse limits) form a dense G(delta)-set. It follows that such inverse limits generated by set-valued functions are generic, and that a generic inverse limit generated by set-valued functions is hereditarily disconnected with no isolated points. (c) 2022 Elsevier B.V. All rights reserved.