On the continuity of global attractors

Let Λ be a complete metric space, and let {S_λ(·) : λ ∈ Λ} be a parametrised family of semigroups with global attractors A_λ. We assume that there exists a fixed bounded set D such that A_λ ⊂ D for every λ ∈ Λ. By viewing the attractors as the limit as t → ∞ of the sets S_λ(t)D, we give simple proofs of the equivalence of ‘equi-attraction’ to continuity (when this convergence is uniform in λ) and show that the attractors A_λ are continuous in λ at a residual set of parameters in the sense of Baire Category (when the convergence is only pointwise).