Useful topologies and separable systems

G. Herden, A. Pallack

Abstract

Let X be an arbitrary set. A topology t on X is said to be useful if every continuous linear preorder on X is representable by a continuous real valued order preserving function. Continuous linear preorders on X are induced by certain families of open subsets of X that are called (linear) separable systems on X. Therefore, in a first step useful topologies on X will be characterized by means of (linear) separable systems on X. Then, in a second step particular topologies on X are studied that do not allow the construction of (linear) separable systems on X that correspond to non representable continuous linear preorders. In this way generalizations of the Eilenberg Debreu theorems which state that second countable or separable and connected topologies on X are useful and of the theorem of Estévez and Hervés which states that a metrizable topology on X is useful, if and only if it is second countable can be proved.

 


Keywords

Complete regular topology; Weak topology; Normal topology; Short topology; Countably bounded topology; Countably bounded linear preorder

Subject classification

54F05; 91B16; 06A05

Full Text:

PDF

Abstract Views

989
Metrics Loading ...

Metrics powered by PLOS ALM


 

Cited-By (articles included in Crossref)

This journal is a Crossref Cited-by Linking member. This list shows the references that citing the article automatically, if there are. For more information about the system please visit Crossref site

1. Utility functions on locally connected spaces
Juan C Candeal, Esteban Induráin, Ghanshyam B Mehta
Journal of Mathematical Economics  vol: 40  issue: 6  first page: 701  year: 2004  
doi: 10.1016/S0304-4068(03)00085-5



Esta revista se publica bajo una licencia de Creative Commons Reconocimiento-NoComercial-SinObraDerivada 4.0 Internacional.

Universitat Politècnica de València

e-ISSN: 1989-4147   https://doi.org/10.4995/agt