One approach to define a topology (instead of open/closed sets) is by the Kuratowski Closure Operator. This operator $cl$ uniquely defines a topology (defined by open/closed sets). But since every topological space has a basis, trivially, I wonder how one may define a basis in terms of the Kuratowski Closure Operator (or its dual, the Interior Operator). Given the fact that any topology can be defined by such an operator, can a basis for a topology be described by e.g. relaxing some of the closure operator axioms?
Background: A topology can be constructed equally by a collection of filters (see my question and the answers here: Characterization of Topology). Since there exists a notion of a filter basis one can construct — based on the equivalence proved in the link — a definition of a basis for a topology in terms of a filter. The resulting basis is known as local basis.
My question is basically wether there a) exists a similar notion for closure/interor operators and if not, b) is it possible to define one?