Abstract
The elaboration of utility theory takes back its source in early economic theory. Despite an obvious practical use for applications and an intuitive appeal, utility theory relies on sophisticated abstract mathematics such as topology. Our interest is primarily on Debreu–Eilenberg’s theorem. On connected separable topological spaces continuous total preorders admit continuous utility representations. We provide a simple proof and derive related results.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Notes
There exists \(\overline{x}, \overline{y} \in X\) such that \(\overline{x} \succ \overline{y}\).
References
Bridges, D., Mehta, G.: Representations of Preferences Orderings. Springer, Berlin (1995)
Debreu, G.: Representation of a preference ordering by a numerical function. In: Thrall, R., Coombs, C., Davis, R. (eds.) Decision Processes. Wiley, New York (1954)
Debreu, G.: Continuity properties of paretian utility. Int. Econ. Rev. 5, 285–293 (1964)
Eilenberg, S.: Ordered topological spaces. Am. J. Math. 63(1), 39–45 (1941)
Fleischer, I.: Numerical representation of utility. J. Soc. Ind. Appl. Math. 9(1), 48–50 (1961)
Herden, G.: On the existence of utility functions. Math. Soc. Sci. 17, 297–313 (1989)
Herden, G.: On the existence of utility functions II. Math. Soc. Sci. 18, 107–117 (1989)
Herden, G.: On some equivalent approaches to mathematical utility theory. Math. Soc. Sci. 29, 19–31 (1995)
Jaffray, J.-Y.: Existence of a continuous utility function: an elementary proof. Econometrica 43, 981–983 (1975)
Mehta, G.: Topological ordered spaces and utility functions. Int. Econ. Rev. 18(3), 779–782 (1977)
Mehta, G.: Some general theorems on the existence of order preserving functions. Math. Soc. Sci. 15, 135–143 (1988)
Nachbin, L.: Topology and Order, Princeton. D. Van Nostrand Company, New Jersey (1965)
Peleg, B.: Utility functions for partially ordered topological spaces. Econometrica 38, 93–96 (1970)
Richter, M.: Continuous and semi-continuous utility. Int. Econ. Rev. 21(2), 293–299 (1980)
Schmeidler, D.: A condition for the completeness of partial preference relations. Econometrica 39(2), 403–404 (1971)
Stigler, G.J.: The development of utility theory. I. J. Polit. Econ. 58(4), 307–327 (1950)
Stigler, G.J.: The development of utility theory. II. J. Polit. Econ. 58(5), 373–396 (1950)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Rébillé, Y. Continuous utility on connected separable topological spaces. Econ Theory Bull 7, 147–153 (2019). https://doi.org/10.1007/s40505-018-0149-4
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s40505-018-0149-4