  The standard model of choice in economics is the maximization of a complete
and transitive preference relation over a fixed set of alternatives. While
completeness of preferences is usually regarded as a strong assumption,
weakening it requires care to ensure that the resulting model still has enough
structure to yield interesting results. This paper takes a step in this
direction by studying the class of "connected preferences", that is,
preferences that may fail to be complete but have connected maximal domains of
comparability. We offer four new results. Theorem 1 identifies a basic
necessary condition for a continuous preference to be connected in the sense
above, while Theorem 2 provides sufficient conditions. Building on the latter,
Theorem 3 characterizes the maximal domains of comparability. Finally, Theorem
4 presents conditions that ensure that maximal domains are arc-connected.
