Useful topologies and separable systems

G. Herden, A. Pallack


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.



Complete regular topology; Weak topology; Normal topology; Short topology; Countably bounded topology; Countably bounded linear preorder

Subject classification

54F05; 91B16; 06A05

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Universitat Politècnica de València

e-ISSN: 1989-4147