Closure theories of powerset theories

A notion of a closure theory of a powerset theory in a ground category is introduced as a generalization of a topology theory of a powerset theory. Using examples of powerset theories in the category Set of sets and in the category of sets with similarity relations, it is proved that these theories can be used as ground theories for closure theories of powerset theories in these two categories. Moreover, it is proved that all these closure theories of powerset theories are topological constructs. A notion of a closure operator which preserves a canonical form of fuzzy objects in these categories is introduced, and it is proved that a closure theory of a powerset theory in the ground category Set is a coreflective subcategory of the closure theory of (Zadeh’s) powerset theory, which preserves canonical forms of fuzzy sets.