On the cardinality of indifference classes
Submitted: 2013-11-26
|Accepted: 2013-11-26
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Keywords:
Indifference potency, Calculable set, Degree of connectedness, Path rank, Separation rank
Supporting agencies:
Abstract:
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.
References:
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).
