  This paper offers a comprehensive treatment of the question as to whether a
binary relation can be consistent (transitive) without being decisive
(complete), or decisive without being consistent, or simultaneously
inconsistent or indecisive, in the presence of a continuity hypothesis that is,
in principle, non-testable. It identifies topological connectedness of the
(choice) set over which the continuous binary relation is defined as being
crucial to this question. Referring to the two-way relationship as the
Eilenberg-Sonnenschein (ES) research program, it presents four synthetic, and
complete, characterizations of connectedness, and its natural extensions; and
two consequences that only stem from it. The six theorems are novel to both the
economic and the mathematical literature: they generalize pioneering results of
Eilenberg (1941), Sonnenschein (1965), Schmeidler (1971) and Sen (1969), and
are relevant to several applied contexts, as well as to ongoing theoretical
work.
