In this paper, using generalized groups and their generalized actions, we define and study the notion of T - spaces. We study properties of the quotient space of a T-space and we present the conditions that imply the Hausdorff property for it. We also prove some essential results about topological generalized groups. As a main result, we show that for each positive integer n there is a topological generalized group T with n identity elements. Moreover, we study the maps between two T -spaces and we consider the notion of T -transitivity.
