Properties of Topologies for the Continuous Representability of All Weakly Continuous Preorders

We investigate properties of strongly useful topologies, i.e., topologies with respect to which every weakly continuous preorder admits a continuous order-preserving function. In particular, we prove that a topology is strongly useful provided that the topology generated by every family of separable...

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Bibliographic Details
Published in:Mathematics (Basel) Vol. 11; no. 20; p. 4335
Main Authors: Bosi, Gianni, Franzoi, Laura, Sbaiz, Gabriele
Format: Journal Article
Language:English
Published: Basel MDPI AG 01.10.2023
Subjects:
Algebraic topology
Continuity (mathematics)
Fuzzy sets
hereditarily separable topology
Lindelöf property
Mathematical functions
Mathematical research
Set theory
strongly useful topology
Topology
Utility functions
weakly continuous preorder
Italy
ISSN:2227-7390, 2227-7390
Online Access:Get full text
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Summary:We investigate properties of strongly useful topologies, i.e., topologies with respect to which every weakly continuous preorder admits a continuous order-preserving function. In particular, we prove that a topology is strongly useful provided that the topology generated by every family of separable systems is countable. Focusing on normal Hausdorff topologies, whose consideration is fully justified and not restrictive at all, we show that strongly useful topologies are hereditarily separable on closed sets, and we identify a simple condition under which the Lindelöf property holds.
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ISSN:2227-7390
2227-7390
DOI:10.3390/math11204335