Closure theories of powerset theories

Abstract

A notion of a closure theory of a powerset theory in a ground categoryis introduced as a generalization of a topology theory of a powerset the-ory. Using examples of powerset theories in the category Set of sets andin the category of sets with similarity relations, it is proved that thesetheories can be used as ground theories for closure theories of powersettheories in these two categories. Moreover, it is proved that all these clo-sure theories of powerset theories are topological constructs. A notion ofa closure operator which preserves a canonical form of fuzzy objects inthese categories is introduced, and it is proved that a closure theory of apowerset theory in the ground category Set is a coreective subcategory ofthe closure theory of (Zadeh's) powerset theory, which preserves canonicalforms of fuzzy sets.