Generic inverse limits with set-valued functions

Abstract

Suppose that for each i≥0, Ii is an interval, and for each i≥1, Pi is a pseudoarc contained in Ii−1×Ii such that πi−1Pi=Ii−1 and πiPi=Ii (πi−1 and πi denote the respective projections of Pi to the intervals Ii−1 and Ii). Then for each i≥1, there is a surjective upper semicontinuous map fi:Ii→2Ii−1 such that the graph Γ(fi)=Pi. We prove that the inverse limit space lim⟵(Ii,fi):={(x0,x1,…)∈Πi=0∞Ii:for eachi≥1,xi−1∈fi(xi)} 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 Gδ-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.