In this paper, we introduce a new class of closed sets which is called αrps –closed sets in topological spaces and we given the relationships of these sets with some other sets. Also, we study some of their properties. Further, will be introduce and study new type of spaces namely T_αrps -space and new type of continuous functions which are ( αrps-continuous functions , αrps- irresolute functions and strongly αrps-continuous functions ) , we introduce several properties of these functions are proved.