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. | Turk J Math (2015) 39: 851 – 863 ¨ ITAK ˙ c TUB ⃝ Turkish Journal of Mathematics doi: Research Article T-Spaces Hassan MALEKI∗, Mohammadreza MOLAEI Department of Pure Mathematics, Shahid Bahonar University of Kerman, Iran Received: • Accepted/Published Online: • Printed: Abstract: 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. Key words: Generalized group, generalized action, transitivity, T -space, quotient space 1. Introduction Groups appear in many branches of mathematics such as number theory, geometry, and the theory of Lie groups, and they can also find many applications even in physics and chemistry. Here we need to recall the notion of topological groups, which play a major role in the geometry and also group representation theory. They, along with their continuous actions, are used to study continuous symmetries, which have many applications in physics and chemistry. Topological groups have both algebraic and topological structures such that the multiplication of the group and inverse function are continuous functions with respect to the topology. More precisely, a topological group G is a group endowed with a topology such that the multiplication mapping m : G × G → G defined by (g, h) → gh and the inversion mapping i : G → G defined by g → g −1 are continuous. For example, any group, endowed with discrete topology or indiscrete topology is a topological group. Many .
