We study the continuity of pullback random attractors \(A_\lambda \), where the parameter belongs to a complete metric space \(\Lambda \). By a Hausdorff sub-norm space, we mean the collection of all nonempty compact subsets of the state space, equipped by the Hausdorff metric as well as a sub-norm. Under weaker conditions, we prove that the binary map \((\lambda , s)\rightarrow A_\lambda (s,\theta _s\omega )\) is continuous at all points of \(\Lambda ^*\times {\mathbb {R}}\) with respect to the Hausdorff metric, where \(\Lambda ^*\) is residual and dense in \(\Lambda \), and that the binary map is continuous under the sub-norm if and only if each fibre of attractors is a point. The proofs of these results are based on the discussion of continuous operators on the Hausdorff sub-norm space as well as the theory of Baire category. For the g-Navier–Stokes equation driven by random density, stochastic noise and time-dependent forces, we establish the residual continuity and full upper semi-continuity of pullback random attractors on the Fréchet space formed from all backward tempered forces.

我们研究了回拉随机吸引子\(A_\lambda \)的连续性，其中参数属于一个完整的度量空间\(\Lambda \)。Hausdorff 子范数空间是指状态空间的所有非空紧致子集的集合，由 Hausdorff 度量以及子范数配备。在较弱的条件下，我们证明了二元映射\((\lambda , s)\rightarrow A_\lambda (s,\theta _s\omega )\)在\(\Lambda ^*\times {\ mathbb {R}}\)关于 Hausdorff 度量，其中\(\Lambda ^*\)是残差且在\(\Lambda \)，并且二值映射在子范数下是连续的，当且仅当每个吸引子纤维都是一个点。这些结果的证明基于对 Hausdorff 子范数空间上的连续算子的讨论以及 Baire 范畴理论。对于由随机密度、随机噪声和时间相关力驱动的 g-Navier-Stokes 方程，我们在由所有反向回火力形成的 Fréchet 空间上建立了回拉随机吸引子的剩余连续性和完全上半连续性。