The non-existence of a universal topological type space

The concept of types was introduced by Harsányi (1967-1968). In the literature there are two approaches for formalizing types, type spaces: the purely measurable and the topological models. In the former framework Heifetz and Samet (1998) showed that the universal type space exists and later Meier (2001) proved that it is complete. In this paper we examine the topological approach and conclude that there is no universal topological type space in the category of topological type spaces.