The paper deals with Alexandroff \(L\)-topological spaces and \(L\)-co-topological spaces, where \((L,I,\ast,\to)\) is a commutative, unital quantale. It is proved that every finite strong \(L\)-co-topological space is Alexandroff and that the category of Alexandroff strong \(L\)-co-topological spaces is the coreflective hull of the subcategory of finite strong \(L\)-co-topological spaces in the category of strong \(L\)-co-topological spaces. However, an example is given to show that there is a finite strong \(L\)-topological space that is not Alexandroff. So, the results illustrate an essential difference between topology and co-topology in the many valued setting.