Continuous-valued social choice

Let Φ be any social choice correspondence satisfying a mild continuity condition in addition to independence of unfeasible alternatives (which requires the correspondence to select the same set of alternatives in two situations if the feasible sets are identical in the two situations and, for each individual, the restriction of the preference ordering to the feasible set is the same) and Arrow's choice axiom (which generalizes transitivity). If the society is finite or countably infinite, then Φ is constant, or dictatorial, or inversely dictatorial, even if the agenda domain is restricted. (The outcome space is normal, connected, and first-countable.).