  Although inverse limits with factor spaces indexed by the positive integers
are most commonly studied, Ingram and Mahavier have defined inverse limits with
set-valued functions broadly enough for any directed index set to be used. In
this paper, we investigate generalized inverse limits whose factor spaces are
indexed by totally ordered sets. Using information about the projections of
such inverse limits onto finitely many coordinates, we generalize various
well-known theorems on connectedness in inverse limits. Moreover, numerous
theorems and examples are given addressing the special case of an inverse limit
with a single idempotent surjective u.s.c. bonding function.
