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

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 = {β ∈ L : β ≺ α}, 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). The grant MTM2006-14925-C02-02 from the Ministry of Education and Science of Spain and FEDER is gratefully acknowledged by the second named author.