Abstract: Using q- open sets, the auther gives the definition to Q_i ( #em/em#= 0, 1, 2, 3, 4) spaces and discuss their characters, and their relationship wi th T_i ( #em/em#= 0, 1, 2, 3, 4) spaces. Topological space X to q- open sets Mani fold { Aα|α∈I } is called q- cover,when only ∪_α∈IAα = X . The auther also defines a new compact space: Q- compact space, and discusses the characters of Q- compact space and the relations between Q- compact space and Q_i- space thus several conclusions are drawn.