Preorderable topologies and order-representability of topological spaces
type: Item
instance of: scholarly article
title: Preorderable topologies and order-representability of topological spaces
zbMATH Open document ID: 1178.54015
author: María Jesús Campión
author: Juan C. Candeal
author: Esteban Induraín
published in: Topology and its Applications
publication date: 2009-11-05T00:00:00Z
review text: A total preorder is called continuous (resp. semicontinuous) with respect to a given topology provided that the order topology (resp. lower or upper topology) induced by the total preorder is coarser than the given topology. Topologies that coincide with the order topology (resp. with the lower topology) induced by a total preorder are called preorderable (resp. lower preorderable). The authors characterize such topologies. They also say that a topology defined on a nonempty set satisfies the continuous representability property (CRP) if every continuous total preorder defined on the given set admits a numerical representation by means of a continuous real-valued order-monomorphism. A topology defined on a nonempty set satisfies the semicontinuous representability property (SRP) if every lower (upper) semicontinuous total preorder defined on the given set admits a numerical representation by means of a lower (upper) semi-continuous real-valued order-monomorphism. Given a topological space \((X,\tau)\) the continuous (semicontinuous) representability property CRP (SRP) on \((X,\tau)\) is characterized by means of the satisfaction of the second countability axiom by every preorderable (lower preorderable) topology on \(X\) that is coarser than \(\tau\).
Mathematics Subject Classification ID: 54F05
Mathematics Subject Classification ID: 06A05
zbMATH DE Number: 5625493
zbMATH Keywords: orderability properties on topological spaces
zbMATH Keywords: topologies induced by a total preorder
zbMATH Keywords: order-representability properties on topological spaces
reviewed by: Hans Peter Künzi
MaRDI profile type: Publication
full work available at URL: j.topol.2009.01.018
OpenAlex ID: W2030329285
cites work: Orderability of topological spaces by continuous preferences.
cites work: Continuous utility functions through scales
cites work: Orderable spaces
cites work: Isotonies on ordered cones through the concept of a decreasing scale
cites work: On the structure of completely useful topologies
cites work: Semicontinuous representability of interval orders on a metrizable topological space
cites work: Representations of preference orderings
cites work: On separation axioms for certain types of ordered topological space
cites work: ON YI'S EXTENSION PROPERTY FOR TOTALLY PREORDERED TOPOLOGICAL SPACES
cites work: Q3595917
cites work: Representable topologies and locally connected spaces
cites work: Topological spaces for which every continuous total preorder can be represented by a continuous utility function
cites work: Useful topologies and separable systems
cites work: Separability, the Countable Chain Condition and the Lindelof Property in Linearly Orderable Spaces
cites work: Linearly Orderable Spaces
cites work: Q4247158
cites work: A topological characterization of ordered spaces
cites work: Orderable topological spaces
DOI: 10.1016/J.TOPOL.2009.01.018
description: scientific article; zbMATH DE number 5625493