Inverse limits with bonding functions whose graphs are arcs

In this paper, we explore connectedness and total disconnectedness of generalized inverse limits on intervals induced by one bonding function f : I → 2 I , which is not necessarily surjective, and whose graph G ( f ) is an arc. As the main results of the paper we prove that (1) if such inverse limit consists of more than one point, then it is infinite, and (2) if such a function f is surjective, then the corresponding inverse limit is never totally disconnected. As a by-product, we obtain a more general result: if for each i, f i denotes a surjective upper semicontinuous bonding function with connected graph, then the resulting generalized inverse limit is never totally disconnected. We also produce examples of such inverse limits (with nonsurjective bonding function f) that are totally disconnected.