Topological properties of spaces ordered by preferences

Abstract

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

Characterization of the existence of maximal elements of acyclic relations

“…The order topology associated with is the topology generated by the subbase S . We have proved in Alcantud [2] that the indifference associated with any asymmetric and negatively transitive binary relation saturates its order topology.…”

“…In some situations, among which we may mention the analysis of intrinsic topologies (cf. Alcantud [2]), they exhibit properties similar to those of linear orders. In this line, one can easily adapt the reasoning given above for linear orders to show that preferences satisfy an analogous property to that enunciated for them.…”

Characterization of the existence of semicontinuous weak utilities for binary relations

“…Definition 5 (See Alcantud 2001, p. 506;Alcantud 1999a, Definition 2.2). Let ≈ be an equivalence relation on a topological space (X, τ ).…”