Generalizing Topological Set Operators

It is well-known that topological spaces can be axiomatically defined by the topological closure operator, i.e., Kuratowski Closure Axioms. Equivalently, they can be axiomatized by other set operators reflecting primitive notions of topology, such as interior operator, derived set operator (or dually, exterior operators, co-derived set operators), or boundary operator. It is also known that topological closure operators (and dually, topological interior operators) can be weakened as in generalized closure (interior) systems. What about boundary operator, exterior operator, and derived set (and co-derived set) operator in the weakened systems? Our paper completely answers this question by showing that these six operators can all be weakened in an appropriate way such that their relationships remain essentially the same as in topological spaces. Our results indicate that topological semantics can be fully relaxed to the weakened systems.