Alexandroff L-co-topological spaces

This article is concerned with Alexandroff L-topological spaces and L-co-topological spaces, where L is a commutative, unital quantale. On one hand, an example is given to show that there is a finite strong L-topological space that is not Alexandroff. On the other hand, 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. So, the results illustrate an essential difference between topology and co-topology in the many valued setting.