Abstract
A topology is said to be strongly useful if every weakly continuous preorder admits a continuous order-preserving function. A strongly useful topology is useful, in the sense that every continuous total preorder admits a continuous utility representation. In this paper, I study the structure of strongly useful topologies. The existence of a natural one-to-one correspondence is proved, between weakly continuous preorders and equivalence classes of families of complete separable systems. In some sense, this result completely clarifies the connections between order theory and topology. Then, I characterize strongly useful topologies and I present a property concerning subspace topologies of strongly useful topological spaces.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Data Availability
No datasets were generated or analysed during the current study.
References
Aliprantis, C.D., Border, K.C.: Infinite Dimensional Analysis - a Hitchhiker’s Guide. Springer, New York (2006)
Bosi, G., Herden, G.: On a strong continuous analogue of the Szpilrajn theorem and its strengthening by Dushnik and Miller. Order 22, 329–342 (2005)
Bosi, G., Herden, G.: On a possible continuous analogue of the Szpilrajn theorem and its strengthening by Dushnik and Miller. Order 23, 271–296 (2006)
Bosi, G., Herden, G.: Continuous multi-utility representations of preorders. J. Math. Econom. 48, 212–218 (2012)
Bosi, G., Herden, G.: On continuous multi-utility representations of semi-closed and closed preorders. Math. Social Sci. 79, 20–29 (2016)
Bosi, G., Herden, G.: The structure of useful topologies. J. Math. Econom. 82, 69–73 (2019)
Bosi, Zuanon, M.: Continuity and continuous multi-utility representations of nontotal preorders: some considerations concerning restrictiveness. In: Bosi, G., Campión, M.J., Candeal, J.C., Indur1áin, E. (eds.), Mathematical Topics on Representations of Ordered Structures. pp. 213–236, Spinger (2020)
Bosi, G., Zuanon, M.: Topologies for the continuous representability of every nontotal weakly continuous preorder. Econom. Theory Bul. 8, 369–378 (2020)
Bosi, G., Zuanon, M.: Topologies for the continuous representability of all continuous total preorders. J. Optim. Theory Appl. 188, 420–431 (2021)
Bosi, G., Estevan, A.: Continuous representations of preferences by means of two continuous functions. J. Optim. Theory Appl. 185, 700–710 (2020)
Campión, M.J., Candeal, J.C., Induráin, E.: The existence of utility functions for weakly continuous preferences on a Banach space. Math. Social Sci. 51, 227–237 (2006)
Campión, M.J., Candeal, J.C., Induráin, E., Mehta, G.B.: Representable topologies and locally connected spaces. Topology and its Applications 154, 2040–2049 (2007)
Campión, M.J., Candeal, J.C., Induráin, E.: Preorderable topologies and order-representability of topological spaces. Topology Appl. 156, 2971–2978 (2009)
Campión, M.J., Candeal, J.C., Induráin, E., Mehta, G.B.: Continuous order representability properties of topological spaces and algebraic structures. J. Korean Math. Soc. 49, 449–473 (2012)
Candeal, J.C., Hervés, C., Induráin, E.: Some results on representation and extension of preferences. J. Math. Econom. 29, 75–81 (1998)
Cigler, J., Reichel, H.C.: Topologie. Bibliographisches Institut, Mannheim-Wien-Zürich (1978)
Engelking, R.: General Topology, Berlin: Heldermann (revised and completed ed.) (1989)
Herden, G.: On the existence of utility functions. Math. Social Sci. 17, 297–313 (1989)
Herden, G.: Topological spaces for which every continuous total preorder can be represented by a continuous utility function. Math. Social Sci. 22, 123–136 (1991)
Herden, G., Pallack, A.: Useful topologies and separable systems. Appl. Gen. Topol. 1, 61–81 (2000)
Herden, G., Pallack, A.: On the continuous analogue of the Szpilrajn theorem I. Math. Social Sci. 43, 115–134 (2002)
Peleg, B.: Utility functions for partially ordered topological spaces. Econometrica 38, 93–96 (1970)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Bosi, G. Continuous Order-Preserving Functions for All Kind of Preorders. Order 40, 87–97 (2023). https://doi.org/10.1007/s11083-022-09598-3
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11083-022-09598-3
Keywords
- Useful topology
- Complete separable system
- Weak topology
- Completely regular space
- Normal space
- Strongly useful topology