Connected generalized inverse limits

Suppose that for each i ⩾ 0 , I i is a closed interval, and for each i ⩾ 1 , f i : I i → 2 I i − 1 is a surjective upper semicontinuous function with a connected graph G i , such that π i − 1 ( G i ) = I i − 1 and π i ( G i ) = I i ( π i − 1 and π i denote the respective projections of G i to the intervals I i − 1 and I i ). We give a condition on the graphs called a C-sequence, and show that { f i : i > 0 } has a C-sequence if and only if there exists a basic open set U = ∏ 0 ⩽ i < n I i × ∏ n ⩽ i ⩽ m U i × ∏ i = m + 1 ∞ I i in ∏ i = 0 ∞ I i containing a closed set A = ∏ 0 ⩽ i < n I i × ∏ n ⩽ i ⩽ m A i × ∏ i = m + 1 ∞ I i , such that lim ← ( I i , f i ) ∩ U = lim ← ( I i , f i ) ∩ A ≠ ∅ , and lim ← ( I i , f i ) ⊄ U .