Connected generalised inverse limits over Hausdorff continua

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