Beside the work of Chichilnisky mentioned by Michael, another interesting use of topology in social choice theory appears in the work of Redekop on Arrow's theorem on economic domains.
- Redekop, J. (1991). Social welfare functions on restricted economic domains. Journal of Economic Theory,
- Redekop, J. (1993a). Arrow-inconsistent economic domains. Social Choice andWelfare,10, 107–126.
- Redekop, J. (1993b). The questionnaire topology on some spaces of economic preferences. Journal of
Mathematical Economics, 22, 479–494.
- Redekop, J. (1993c). Social welfare functions on parametric domains. Social Choice andWelfare,10, 127–148.
- Redekop, J. (1995). Arrow theorems in economic environments. In W. A. Barnett, H. Moulin, M. Salles,
& N. J. Schofield (Eds.), Social choice,welfare,and ethics (pp. 163–185). Cambridge: Cambridge University
- Redekop, J. (1996). Arrow theorems in mixed goods, stochastic, and dynamic economic environments.
Social Choice and Welfare,13, 95–112.
Arrow's impossibility theorem was originally proven for an abstract set of alternative, allowing every possible preference profile over this set of alternatives. The question that Redekop (and others) asked was : is there an equivalent of Arrow's theorem when the alternatives are bundles of goods, and agent have "classical" preferences over those goods (monotonic, convex, continuous, selfish,...).
More precisely, the question was whether there would exists a social welfare function satisfying the three Arrovian axioms (Independence of Irrelevant Alternative, Weak Pareto and Non-Dictatorship) on these Economic domains (see
Le Breton, Michel, and John A. Weymark. "Chapter Seventeen-Arrovian Social Choice Theory on Economic Domains." Handbook of social choice and welfare 2 (2011): 191-299 for a great review, which this answer is based upon).
Roughly, Redekop's work shows that, for some of those economic problems, if a domain of preferences admits an Arrovian social welfare function, the domain must be "small" in some topological sense.
For instance, in Redekop (1991), he introduces an ingenious topology on sets of preferences he dubbed the questionnaire topology, and shows that, in a public goods economy, if a domain of preferences admits an Arrovian social welfare function, then the domain must be nowhere dense according to this topology (i.e. the closure of the domain contains no open set).