Applications of localic separation axioms, compactness axioms, representations, and compactifications to poslat topological spaces

Abstract

Fuzzy topology first appeared about twenty-seven years ago with Chang's initial paper (1968). Yet the fundamental issues of compactness and low-order separation axioms remain unresolved. This paper resolves these issues with respect to topological compactification, categorical compactification, and Stone representation theorems, using the theory of locales and previous work of the author in fuzzy sobriety. Our results give the first compactification reflectors for all traditional fuzzy topology, construct classes of generalized Stone representations of Boolean algebras and distributive lattices, and show categorically that the fuzzy analogue of classical compact Hausdorff spaces is the categories of Chang-compact, localic regular, sober fuzzy topological spaces.