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

A lower limit function f _* of a given map f from a topological space X to a complete lattice L is defined by f _* (x) = sup{ inf f(U) : U in U(x) } for all x in X. Let \GammaL = (L, \gamma(L)) with \gamma(L) a topology on L. We discuss conditions on \GammaL under which C(X, \GammaL) becomes a fuzzy topology on X and (·) _* : 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

(*)   \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 (*), which thus provides a short proof that each completely distributive lattice is hypercontinuous (hence continuous).

Date received: June 24, 1996

Copyright © 1996 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caah-59.