Connected generalized inverse limits

Suppose that for each i⩾0, Ii is a closed interval, and for each i⩾1, fi:Ii→Ii−12 is a surjective upper semicontinuous function with a connected graph Gi, such that πi−1(Gi)=Ii−1 and πi(Gi)=Ii (πi−1 and πi denote the respective projections of Gi to the intervals Ii−1 and Ii). We give a condition on the graphs called a C-sequence, and show that {fi:i>0} has a C-sequence if and only if there exists a basic open set in containing a closed set , such that , and .