On fuzzy monotone convergence [formula omitted]-cotopological spaces

Abstract

In this paper, we generalize the concept of a monotone convergence space (also called a d-space) to the setting of a Q-cotopological space, where Q is a commutative and integral quantale. We establish a D-completion for every stratified Q-cotopological space, which is a category reflection of the category SQ-CTop of stratified Q-cotopological spaces onto the full subcategory SQ-DCTop of monotone convergence Q-cotopological spaces. By introducing the notion of a tapered set, a direct characterization of the completion is obtained: the D-completion of each stratified Q-cotopological space X consists exactly of those tapered closed sets in X. We show that the D-completion can be applied to obtain a universal fuzzy directed completion of a Q-ordered set by endowing it with the Scott cotopology, taking the D-completion, and then passing to the specialization Q-order. Consequently, the category Q-DOrd of fuzzy directed complete Q-ordered sets and Scott continuous functions is reflective in the category Q-Ordσ of Q-ordered sets and Scott continuous functions.