Topologies for the continuous representability of every nontotal weakly continuous preorder
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Topologies for the continuous representability of every nontotal weakly continuous preorder Gianni Bosi1 · Magalì Zuanon2 Received: 31 May 2020 / Accepted: 2 July 2020 © Society for the Advancement of Economic Theory 2020
Abstract Necessary and sufficient conditions on a topology t on an arbitrary set X are presented, under which every not necessarily total preorder, which in addition satisfies a general continuity condition, namely weak continuity, admits a continuous orderpreserving real-valued function. Some interesting properties associated with this notion are studied. Keywords Complete separable system · Useful topology · Strongly useful topology · Normal Hausdorff-space JEL classification C02 · D00
1 Introduction Let (X, t) be a topological space. Then, in order for a preorder ≾ on (X, t) to being representable by a continuous order-preserving real-valued function, it is necessary that for every pair (x, y) ∈≺ there exists a continuous increasing real-valued function uxy ∶ (X, ≾, t) → (ℝ, ≤, tnat ) such that uxy (x) < uxy (y) . Therefore, in this paper a preorder ≾ on (X, t) is defined to be weakly continuous if it satisfies the just defined monotony behaviour. The concept of weak continuity of a preorder ≾ on a topological space (X, t), which was introduced by Herden and Pallack (2000) [see also Bosi and Herden (2005, 2006)], is equivalent to requiring, for every pair (x, y) ∈≺ , to exist a decreasing complete separable system E on (X, t) such that there exist sets * Gianni Bosi [email protected] Magalì Zuanon [email protected] 1
Dipartimento di Scienze Economiche, Aziendali, Matematiche e Statistiche, Università di Trieste, via Università 1, Trieste 34123, Italy
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Dipartimento di Economia e Management, Università degli Studi di Brescia, Contrada Santa Chiara 50, Brescia 25122, Italy
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G. Bosi, M. Zuanon
E ⊂ E ⊂ E′ in E such that x ∈ E and y ∉ E� (see Bosi and Herden (2019) [Definition 2.1 and Definition 2.2]). In this paper, we are interested in studying conditions on a topology t on a set X, according to which every weakly continuous preorder on the topological space (X, t) admits a continuous order-preserving real-valued function u (i.e., a continuous increasing function u ∶ (X, ≾, t) → (ℝ, ≤, tnat ) such that u(x) < u(y) for all pairs (x, y) ∈≺ ). Therefore, a topology t on X is said to be strongly useful if every weakly continuous preorder ≾ on (X, t) admits a continuous orderpreserving real-valued function. It should be noted that the concept of a strongly useful topology includes the consideration of preorders which are not total. The importance of considering nontotal preorders as more realistic representations of individual preferences has been recognized long ago in the seminal papers of Aumann (1962) and Peleg (1970). It is immediate to check that a strongly useful topology t on X is also useful, i.e., every continuous total preorder ≾ on (X, t) is representable by a continuous utility function u. We recall that the concept of a useful topol
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