Connected generalised inverse limits over Hausdorff continua

Abstract

Suppose that for each i⩾0, Xi is a Hausdorff continuum, and fi+1:Xi+1→2Xi is an upper semicontinuous function with a connected graph Gi+1, such that πi(Gi+1)=Xi and πi+1(Gi+1)=Xi+1 (πi and πi+1 denote the respective projections of Gi+1 to Xi and Xi+1). We give a condition on the graphs called an HC-sequence, and show that {fi:i>0} admits an HC-sequence if and only if there exists a connected basic open set U=∏0⩽i<mXi×∏m⩽i⩽nUi×∏i>nXi in ∏i∈NXi containing a closed set A=∏0⩽i<mXi×∏m⩽i⩽nAi×∏i>nXi, such that lim←(Xi,fi)∩U=▪(Xi,fi)∩A≠∅, and ▪(Xi,fi)⊄U. An immediate corollary of this is that if the graphs admit an HC-sequence then ▪(Xi,fi) is disconnected. We give a theorem analogous to the Subsequence Theorem where we define a generalised inverse limit ▪(Yi,gi) such that each of the functions gi is obtained from a finite subsequence of 〈fi:i∈N〉, and show that ▪(Yi,gi) is homeomorphic to ▪(Xi,fi).