According to the definition of quasiweak almost periodic point, we introduce the definition of QW_G(f) in the paper. We studied topological structure of G-quasiweak almost periodic point set. We have the following two results: (1) Let (Y,d^′) be a compact G–metric space and f : X → X be equicontinuous. Let {f_n}^∞_n=1 be G-strongly uniform converge to the map f and x_k ∈ QW_G(f_n). If lim_{k → ∞} x_k = x, then x ∈ QW_G(f); (2) Let (Y,d^′) be a compact G-metric space and f : X → X be equicontinuous. If {f_n}^∞_n=1 is G-strongly uniform converge to f, then we have lim sup QW_G(f_n) ⊂ QW_G(f). The conclusions results generalize the corresponding results given in [Journal of Southwest China Normal University: Natural Science Edition 44(2019), 40-44].