Connectedness and inverse limits with set-valued functions on intervals

In [4] it was shown that an inverse limit with upper semicontinuous set-valued functions on intervals is connected if and only if it does not admit a CC-sequence. We introduce a related notion, a component base. Using this new tool we demonstrate how proofs of connectedness theorems can be simplified, and we prove a number of new results on connectedness. In particular, results relating to the size and number of components.