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, Volume 25, Issue 1, pp 9–17 | Cite as

Approximating Orders in Meet-Continuous Lattices and Regularity Axioms in Many Valued Topology

  • Ulrich Höhle
  • Tomasz Kubiak
Article

Abstract

It is shown that in a meet-continuous lattice L endowed with a multiplicative auxiliary order ≺ the family of all members of L which satisfy the axiom of approximation, i.e. α = \(\bigvee\){βL : βα}, is closed under finite infs and arbitrary sups. This is a key ingredient of a meet-continuous lattice proof that both regularity and complete regularity of many valued topology have subbasic characterizations. As a consequence, the frame law can now be eliminated from some fundamental results on completely regular L-valued topological spaces (e.g., this is the case in regard to the Tychonoff embedding theorem).

Keywords

Meet-continuous lattices Multiplicative auxiliary order L-valued topology Regularity Complete regularity L-cube 

Mathematics Subject Classifications (2000)

06B23 54A40 54D10 

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Copyright information

© Springer Science+Business Media B.V. 2007

Authors and Affiliations

  1. 1.Fachbereich C Mathematik und NaturwissenschaftenBergische UniversitätWuppertalGermany
  2. 2.Wydział Matematyki i InformatykiUniwersytet im. Adama MickiewiczaPoznańPoland

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