{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,12,18]],"date-time":"2025-12-18T14:29:43Z","timestamp":1766068183754,"version":"build-2065373602"},"reference-count":20,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2025,3,3]],"date-time":"2025-03-03T00:00:00Z","timestamp":1740960000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>In this paper, we deal with continuous multi-utility representations for closed preorders. We introduce the definition of a net-compact topology, which generalizes the concept of a sequentially compact topology. Indeed, a sequentially compact and first countable topological space is net-compact. First, we show that if every closed preorder on a net-compact Hausdorff topological space has a continuous multi-utility representation, then the topology is normal. Second, we prove that every closed preorder on a normal and net-compact Hausdorff topological space admits a continuous multi-utility representation.<\/jats:p>","DOI":"10.3390\/axioms14030188","type":"journal-article","created":{"date-parts":[[2025,3,3]],"date-time":"2025-03-03T09:04:49Z","timestamp":1740992689000},"page":"188","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Net-Compact Hausdorff Topologies and Continuous Multi-Utility Representations for Closed Preorders"],"prefix":"10.3390","volume":"14","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-3799-4630","authenticated-orcid":false,"given":"Gianni","family":"Bosi","sequence":"first","affiliation":[{"name":"DEAMS, Universit\u00e0 di Trieste, 34127 Trieste, Italy"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-3500-6843","authenticated-orcid":false,"given":"Gabriele","family":"Sbaiz","sequence":"additional","affiliation":[{"name":"DEAMS, Universit\u00e0 di Trieste, 34127 Trieste, Italy"}]},{"given":"Magal\u00ec","family":"Zuanon","sequence":"additional","affiliation":[{"name":"DEM, Universit\u00e0 di Brescia, 25123 Brescia, Italy"}]}],"member":"1968","published-online":{"date-parts":[[2025,3,3]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"554","DOI":"10.1016\/j.jmateco.2011.07.003","article-title":"On the multi-utility representation of preference relations","volume":"47","author":"Evren","year":"2011","journal-title":"J. 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Dokl."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"141","DOI":"10.1017\/S0004972700004597","article-title":"Existence of an order-preserving function on a normally preordered space","volume":"34","author":"Mehta","year":"1986","journal-title":"Bull. Aust. Math. Soc."},{"unstructured":"Nachbin, L. (1965). Topology and Order, Van Nostrand.","key":"ref_14"},{"unstructured":"Bridges, D.S., and Mehta, G.B. (2013). Representations of Preferences Orderings, Springer Science & Business Media.","key":"ref_15"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"445","DOI":"10.2307\/1909888","article-title":"Utility Theory Without the Completeness Axiom","volume":"30","author":"Aumann","year":"1962","journal-title":"Econometrica"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"523","DOI":"10.2140\/pjm.1971.38.523","article-title":"Bicontinuous preordered topological spaces","volume":"38","author":"McCartan","year":"1971","journal-title":"Pacific J. 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