Useful topologies and separable systems

• Herden, G. ^[1] ; Pallack, A. ^[1]
  1. [1] Universität/GH Essen
• Localización: Applied general topology, ISSN-e 1989-4147, ISSN 1576-9402, Vol. 1, Nº. 1, 2000, págs. 61-81
• Idioma: inglés
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• Resumen

  • 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.