Open AccessUseful topologies and separable systems
Author(s) -
Gerhard Herden,
Andreas Pallack
Publication year - 2000
Publication title -
applied general topology
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.638
H-Index - 13
eISSN - 1989-4147
pISSN - 1576-9402
DOI - 10.4995/agt.2000.3024
Subject(s) - separable space , mathematics , metrization theorem , preorder , countable set , network topology , discrete mathematics , function (biology) , order (exchange) , topology (electrical circuits) , combinatorics , computer science , mathematical analysis , finance , evolutionary biology , economics , biology , operating system
Let X be an arbitrary set. A topology t on X is said to be useful if every continuous linear preorder on X is representable by a continuous real valued order preserving function. Continuous linear preorders on X are induced by certain families of open subsets of X that are called (linear) separable systems on X. Therefore, in a first step useful topologies on X will be characterized by means of (linear) separable systems on X. Then, in a second step particular topologies on X are studied that do not allow the construction of (linear) separable systems on X that correspond to non representable continuous linear preorders. In this way generalizations of the Eilenberg Debreu theorems which state that second countable or separable and connected topologies on X are useful and of the theorem of Estévez and Hervés which states that a metrizable topology on X is useful, if and only if it is second countable can be proved