In this paper, the concepts of the base and subbase in intuitionistic Ifuzzy topological spaces are introduced, and use them to discuss fuzzy continuous mapping and fuzzy open mapping. We also study the base and subbase in the product of intuitionistic I-fuzzy topological spaces, and T2 separation in product intuitionistic I-fuzzy topological spaces. Finally, the relation between the generated product intuitionistic Ifuzzy topological spaces and the product generated intuitionistic Ifuzzy topological spaces are studied.