On the existence of utility functions

Herden, Gerhard

doi:10.1016/0165-4896(89)90058-9

Cited by 67 publications

(47 citation statements)

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“…For POTS, some results have been obtained, e.g., by Mehta, who provided characterizations of separability and second countability in [16], and by Herden, who gave a characterization of second countability in [11]. Here, we show that GPO-spaces, which are not T 0 unless the order is linear, are (hereditarily) normal and (completely) regular.…”

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confidence: 72%

Topological properties of spaces ordered by preferences

Alcantud

1999

International Journal of Mathematics and Mathematical Sciences

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Abstract.In this paper, we analyze the main topological properties of a relevant class of topologies associated with spaces ordered by preferences (asymmetric, negatively transitive binary relations). This class consists of certain continuous topologies which include the order topology. The concept of saturated identification is introduced in order to provide a natural proof of the fact that all these spaces possess topological properties analogous to those of linearly ordered topological spaces, inter alia monotone and hereditary normality, and complete regularity.Keywords and phrases. Saturated identifications, order topology, GPO-spaces, POTS, monotonically normal, normal, regular.1991 Mathematics Subject Classification. 06F30, 54F05.1. Introduction. This paper is devoted to perform a thorough study of the topological properties of spaces ordered by preferences (asymmetric, negatively transitive binary relations). The general problem of studying the topologies associated with ordered spaces, among which the most popular are the order and the interval topologies, has proved to be rather difficult and, thus, the literature abounds with results on the behavior of such topologies (cf. Erné [6], Kolibiar [12], Nachbin [17], Northam [18]). However, the particular case of linear orders has been deeply analyzed and it has been shown that they possess a rich topological structure. We prove that spaces ordered by preferences display most of the topological properties of linearly ordered spaces. Besides, in our study, we also consider separability and the axioms of countability.Let us recall the main results in this line which concern linear orders. Any linearly ordered topological space (abbreviated: LOTS) is T 1 and hereditarily normal. See, e.g., Steen [13]. This class of topological spaces or, in general, those whose topology is related in some sense to a linear order, have been widely investigated (cf. Birkhöff . A particularly interesting case is that of GO-spaces (which stands for Generalized Ordered spaces) because they include LOTS and satisfy properties analogous to theirs. In fact, a remarkable result by R. W. Heath and D. J. Lutzer in [10] shows that every GO-space is T 1 and monotonically normal, which is an improvement of the aforementioned result that LOTS are hereditarily normal. On the other hand, van Dalen and Wattel [20] characterized those topological spaces that are LOTS or GO-spaces. The monographies of Faber [7] and van Wouwe [21] are devoted to the study of the topological properties of GO-spaces, and especially to metrizability.In this paper, we define and analyze the natural generalization of the concepts of

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confidence: 72%

Topological properties of spaces ordered by preferences

Alcantud

1999

International Journal of Mathematics and Mathematical Sciences

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“…The property introduced in Part (iv) of the statement of Theorem 1 is called separability in the sense of Jaffray, or separability in the sense of Herden (see [57,58]). …”

Section: Remarkmentioning

confidence: 99%

A Survey on the Mathematical Foundations of Axiomatic Entropy: Representability and Orderings

Campión

Gómez‐Polo

Induráin

et al. 2018

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Different abstract versions of entropy, encountered in science, are interpreted in the light of numerical representations of several ordered structures, as total-preorders, interval-orders and semiorders. Intransitivities, other aspects of entropy as competitive systems, additivity, etc., are also viewed in terms of representability of algebraic structures endowed with some compatible ordering. A particular attention is paid to the problem of the construction of an entropy function or their mathematical equivalents. Multidisciplinary comparisons to other similar frameworks are also discussed, pointing out the mathematical foundations.

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“…Classical works on this subject are those by Eilenberg [5], Nachbin [12], and Fishburn [6]. Further studies on the general case of preordered sets appear in Mehta [9, 10] and Herden [7,8].…”

Section: Introductionmentioning

confidence: 99%