The order relations between closure operators and topped intersection structures on the same set are discussed. Some partial orders on C(X) (the set of all closure operators on X)、I(X) (the set of all Topped intersection-structures on X)、AC(X)(the set of all algebraic closure operators on X) and AI(X)(the set of all algebraic closure operators on X) are introduced, respectively. These four sets are proved to be complete lattices.It is proved that C(X) is dually isomorphic to I(X), and AC(X) is dually isomorphic to AI(X). TC(X) (the set of all topological closure operators on X) is dually isomorphic to TI(X) (the set of all topological intersection structures on X), and MC(X) (the set of all matroid closure operators on X) is dually isomorphic to MI(X) (the set of all matroid intersection structures on X).