Connected generalised inverse limits over Hausdorff continua

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⩽inXi in ∏i∈NXi containing a closed set A=∏0⩽inXi, such that , and . An immediate corollary of this is that if the graphs admit an HC-sequence then is disconnected. We give a theorem analogous to the Subsequence Theorem where we define a generalised inverse limit such that each of the functions gi is obtained from a finite subsequence of 〈fi:i∈N〉, and show that is homeomorphic to .