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| Main Authors: | , , |
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| Format: | Preprint |
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2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2401.13392 |
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| _version_ | 1866913207400005632 |
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| author | Bosi, Gianni Estevan, Asier Raventós, Armajac |
| author_facet | Bosi, Gianni Estevan, Asier Raventós, Armajac |
| contents | In the present paper we study necessary and sufficient conditions for the existence of a semicontinuous and finite Richter-Peleg multi-utility for a preorder. It is well know that, given a preorder on a topological space, if there is a lower (upper) semicontinuous Richter-Peleg multi-utility, then the topology of the space must be finer than the Upper (resp. Lower) topology. However, this condition does not guarantee the existence of a semicontinuous representation.
We search for finer topologies which are necessary for semicontinuity, as well as that they could guarantee the existence of a semicontinuous representation. As a result, we prove that Scott topology (that refines the Upper one) must be contained in the topology of the space in case there exists a finite lower semicontinuous Richter-Peleg multi-utility. However, as it is shown, the existence of this representation cannot be guaranteed. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_13392 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A topological study for the existence of lower-semicontinuous Richter-Peleg multi-utilities Bosi, Gianni Estevan, Asier Raventós, Armajac General Topology In the present paper we study necessary and sufficient conditions for the existence of a semicontinuous and finite Richter-Peleg multi-utility for a preorder. It is well know that, given a preorder on a topological space, if there is a lower (upper) semicontinuous Richter-Peleg multi-utility, then the topology of the space must be finer than the Upper (resp. Lower) topology. However, this condition does not guarantee the existence of a semicontinuous representation. We search for finer topologies which are necessary for semicontinuity, as well as that they could guarantee the existence of a semicontinuous representation. As a result, we prove that Scott topology (that refines the Upper one) must be contained in the topology of the space in case there exists a finite lower semicontinuous Richter-Peleg multi-utility. However, as it is shown, the existence of this representation cannot be guaranteed. |
| title | A topological study for the existence of lower-semicontinuous Richter-Peleg multi-utilities |
| topic | General Topology |
| url | https://arxiv.org/abs/2401.13392 |