Examples of single named sets give us usual sets. Really in the framework of named set theory we can have a new understanding of different trends in the mathematical set theory. The first fact that lies on the surface is that any usual set is a single named set because we cannot speak about, use or construct any set without giving a name to it. However, this name always exists. The most general kind is a topological manifold.
Categories play an important role in modern mathematics.
Any category K consists of two classes Ob K of objects from K and Mor K of morphisms from K . A partial binary composition of morphisms is defined in Mor K . By this correspondence, any functor between abstract categories is mapped into a homomorphism of categorical systems of named sets. If we take mathematical logic we also see a lot of different named sets. As an example we can take such important construction as model.