Useful topologies and separable systems


  • G. Herden Universität/GH Essen
  • A. Pallack Universität/GH Essen



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


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.



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How to Cite

G. Herden and A. Pallack, “Useful topologies and separable systems”, Appl. Gen. Topol., vol. 1, no. 1, pp. 61–81, Oct. 2000.



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