Characterization of Useful Topologies in Mathematical Utility Theory by Countable Chain Conditions
Under the additional assumption of complete regularity, we furnish a simple characterization of all the topologies such that every continuous total preorder is representable by a continuous utility function. In particular, we prove that a completely regular topology satisfies such property if, and o...
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| Format: | Article |
| Language: | English |
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MDPI AG
2022-04-01
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| Series: | Axioms |
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| Online Access: | https://www.mdpi.com/2075-1680/11/5/195 |
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| author | Gianni Bosi Magalì Zuanon |
| author_facet | Gianni Bosi Magalì Zuanon |
| author_sort | Gianni Bosi |
| collection | DOAJ |
| description | Under the additional assumption of complete regularity, we furnish a simple characterization of all the topologies such that every continuous total preorder is representable by a continuous utility function. In particular, we prove that a completely regular topology satisfies such property if, and only if, it is separable and every linearly ordered collection of clopen sets is countable. Since it is not restrictive to refer to completely regular topologies when dealing with this kind of problem, this is, as far as we are concerned, the simplest characterization of this sort available in the literature. All the famous utility representation theorems are corollaries of our result. |
| first_indexed | 2024-03-10T03:20:18Z |
| format | Article |
| id | doaj.art-475a5343e25a4cf48f7ac4a313b12bc4 |
| institution | Directory Open Access Journal |
| issn | 2075-1680 |
| language | English |
| last_indexed | 2024-03-10T03:20:18Z |
| publishDate | 2022-04-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Axioms |
| spelling | doaj.art-475a5343e25a4cf48f7ac4a313b12bc42023-11-23T10:03:55ZengMDPI AGAxioms2075-16802022-04-0111519510.3390/axioms11050195Characterization of Useful Topologies in Mathematical Utility Theory by Countable Chain ConditionsGianni Bosi0Magalì Zuanon1DEAMS, Università di Trieste, Via Valerio 4/1, 34127 Trieste, ItalyDEM, Università di Brescia, 25121 Brescia, ItalyUnder the additional assumption of complete regularity, we furnish a simple characterization of all the topologies such that every continuous total preorder is representable by a continuous utility function. In particular, we prove that a completely regular topology satisfies such property if, and only if, it is separable and every linearly ordered collection of clopen sets is countable. Since it is not restrictive to refer to completely regular topologies when dealing with this kind of problem, this is, as far as we are concerned, the simplest characterization of this sort available in the literature. All the famous utility representation theorems are corollaries of our result.https://www.mdpi.com/2075-1680/11/5/195continuous utilityuseful topologychaincomplete regularity |
| spellingShingle | Gianni Bosi Magalì Zuanon Characterization of Useful Topologies in Mathematical Utility Theory by Countable Chain Conditions Axioms continuous utility useful topology chain complete regularity |
| title | Characterization of Useful Topologies in Mathematical Utility Theory by Countable Chain Conditions |
| title_full | Characterization of Useful Topologies in Mathematical Utility Theory by Countable Chain Conditions |
| title_fullStr | Characterization of Useful Topologies in Mathematical Utility Theory by Countable Chain Conditions |
| title_full_unstemmed | Characterization of Useful Topologies in Mathematical Utility Theory by Countable Chain Conditions |
| title_short | Characterization of Useful Topologies in Mathematical Utility Theory by Countable Chain Conditions |
| title_sort | characterization of useful topologies in mathematical utility theory by countable chain conditions |
| topic | continuous utility useful topology chain complete regularity |
| url | https://www.mdpi.com/2075-1680/11/5/195 |
| work_keys_str_mv | AT giannibosi characterizationofusefultopologiesinmathematicalutilitytheorybycountablechainconditions AT magalizuanon characterizationofusefultopologiesinmathematicalutilitytheorybycountablechainconditions |