The questionnaire topology on some spaces of economic preferences
We define and characterize two well-known topologies on spaces of preferences, namely the Kannai and closed-convergence topologies. We show that on spaces of economic preferences (e.g. continuous and monotonic preferences on \(\mathbb{R}^ l_ +\)), both topologies coincide and have an especially simple characterization. Namely, they agree with the questionnaire topology, which takes as a base the set of questionnaire sets \(Q((x_ 1,y_ 1),\dots(x_ n,y_ n))=\bigcap^ n_{i=1}\{p| x_ i\widehat py_ i\}\), where \(\widehat p\) denotes strict preference. The characterization permits the statement and proof of some theorems on the innocuousness of the assumption of continuity, given monotonicity or convexity; and it allows for dramatically simplified proofs of some denseness results resembling those of \textit{Y. Kannai} [J. Math. Econ. 1, 101-106 (1974; Zbl 0288.90011)], \textit{A. Mas- Colell} [J. Econ. Theory 8, 305-320 (1974)] and \textit{A. Araujo} [J. Math. Econ. 14, 61-66 (1985; Zbl 0582.90013)].
scientific article; zbMATH DE number 458841