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Properties of Topologies for the Continuous Representability of All Weakly Continuous Preorders
Version 1
: Received: 18 September 2023 / Approved: 19 September 2023 / Online: 19 September 2023 (03:49:56 CEST)
A peer-reviewed article of this Preprint also exists.
Bosi, G.; Franzoi, L.; Sbaiz, G. Properties of Topologies for the Continuous Representability of All Weakly Continuous Preorders. Mathematics 2023, 11, 4335. Bosi, G.; Franzoi, L.; Sbaiz, G. Properties of Topologies for the Continuous Representability of All Weakly Continuous Preorders. Mathematics 2023, 11, 4335.
Abstract
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.
Keywords
strongly useful topology; weakly continuous preorder; hereditarily separable topology; Lindelöf property
Subject
Computer Science and Mathematics, Geometry and Topology
Copyright: This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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