On the cardinality of indifference classes
Keywords:Indifference potency, Calculable set, Degree of connectedness, Path rank, Separation rank
Let “≤” be a continuous total preorder on some topological space (X, t). Then the cardinality or at least lower and upper bounds of the cardinality of the indifference (equivalence) classes of “≤ ” will be computed. In addition, the relevance of these bounds in mathematical utility theory and the theory of orderable topological spaces will be discussed.
K. Arrow and F. Hahn, General competetive analysis, (Oliver and Boyd, Edinburgh, 1971).
A.F. Beardon, Totally ordered subsets of Euclidean space, J. Math. Econom. 23 (1994), 391-393. http://dx.doi.org/10.1016/0304-4068(94)90022-1
T. Bewley, Existence of equilibria in economics with infinitely many commodities, J. of Econom. Theory 4 (1972), 514-540. http://dx.doi.org/10.1016/0022-0531(72)90136-6
G. Birkhoff, Lattice Theory, (American Mathematical Society, Rhode Island, 1940 (First edition), 1967 (Third edition)).
D. Bridges and G.B. Mehta, Representation of preference orderings, (Springer, New York, 1995). http://dx.doi.org/10.1007/978-3-642-51495-1
J.C. Candeal and E. Induráin, Utility functions on chains, J. Math. Econom. 22 (1993), 161-168. http://dx.doi.org/10.1016/0304-4068(93)90045-M
J.C. Candeal, E. Induráin and G.B. Mehta, Further remarks on totally ordered representable subsets of Euclidean space, J. Math. Econom. 25 (1996), 381-390. http://dx.doi.org/10.1016/0304-4068(95)00734-2
J. Dugundji, Topology, (Allyn and Bacon, Boston, 1966).
S. Eilenberg, Ordered topological spaces, Amer. J. Math. 24 (1941), 305-309.
G. Herden, On the semicontinuous and continuous analogue of the Szpilrajn theorem, in preparation, University of Essen, 2003.
L. Jones, A competetive model of product differentiation, Econometrica 52 (1984), 507-530. http://dx.doi.org/10.2307/1911501
H. Kok, Connected orderable spaces, (Mathematical Centre Tracts, Am-sterdam, 1973).
A. Mas-Colell, The price equilibrium existence problem in topological vector lattices, Econometrica 54 (1986), 1039-1053. http://dx.doi.org/10.2307/1912321
G.B. Mehta, Infinite dimensional Arrow-Hahn theorem, Preprint, University of Brisbane, 1989.
S. Willard, General topology, (Addison-Wesley, Mass.-London-Don Mills, 1970).
How to Cite
This journal is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.