The Eighth Prague Topological Symposium August 18-24, 1996 Economical University Prague, Czech Republic

Lower Limits of Lattice-Valued Functions and the Associated Fuzzy Topologies by Tomasz Kubiak ( )

A lower limit function f_\star of a given map f from a topological space X to a complete lattice L is defined by f_\star(x) = {\sup}{{\inf} f(U) : U in {\Cal U} (x)} for all x in X . Let \Gamma L = (L, \gamma (L)) with \gamma(L) a topology on L . We discuss conditions on \Gamma L under which C(X,\Gamma L) becomes a fuzzy topology on X and (\cdot)_\star : L^X --> L^X is the associated interior operator. For L a meet-continuous lattice with \gamma (L) weaker than the Scott topology of L , this is the case if and only if \hfill\break (*) \qquad \alpha=\sup{\beta in L : \alpha in Int_\gamma(L)\uparrow\beta} for every \alpha in L .

There is an easy argument showing that every completely distributive lattice L with \gamma (L) stronger than the upper topology satisfies the condition (\star) , which thus provides a short proof that each completely distributive lattice is hypercontinuous (hence continuous).