Abstract: In this paper, a new topological space as well as minimal families topological space is established on completely distributive lattices by using the results of the minimal families, the result that the largest standard minimal families are only nontrivial open sets on this topology is proved.The properties and the sufficient and necessary condition for functional continuity on the minimal families topological space shall be given, the relation between the continuity and the generalized order homomorphism under the new topological space is discussed, the result that the standard minimal families mapping on completely distributive lattices is the generalized order homomorphism is obtained.