Infra topology has efficient tools of knowledge discovery. It has many applications in graph theory, fractals and game theory. One of the aims of this paper is to study some near infra open sets in infra topological spaces. Additionally, some results for near infra open and closed sets are studied. In fact, the union of an infra preclosed set and an infra α-closed set is an infra preclosed set. While, the intersection of an infra α-open set and an infra β-open set is infra β-open. Moreover, infra β-continuity is introduced and the relationships between some types of infra continuous functions is discussed. A comparison between near infra open sets and some other types of near open sets is investigated. Also, A comparison between near infra open sets and some other types of near open sets is obtained. Finally, we introduce two examples on infra topology to satisfy the relationships between infra β-continuity and some other types of infra continuity.