1. GlobalattractorsThe global attractor of a dynamical system is the unique compact invariant setthat attracts the trajectories starting in any bounded set at a uniform rate. Intro-duced by Billotti & LaSalle [3], they have been the subject of much research sincethe mid-1980s, and form the central topic of a number of monographs, includingBabin & Vishik [1], Hale [9], Ladyzhenskaya [13], Robinson [16], and Temam [18].The standard theory incorporates existence results [3], upper semicontinuity [10],and bounds on the attractordimension [7]. Global attractorsexist for many infinite-dimensional models [18], with familiar low-dimensional ODE models such as theLorenz equations providing a testing ground for the general theory [8].While upper semicontinuity with respect to perturbations is easy to prove, lowersemicontinuity (and hence full continuity) is more delicate, requiring structuralassumptions on the attractor or the assumption of a uniform attraction rate. How-ever, Babin & Pilyugin [2] proved that the global attractor of a parametrised setof semigroups is continuous at a residual set of parameters, by taking advantage ofthe known upper semicontinuity and then using the fact that upper semicontinuousfunctions are continuous on a residual set.Here we reprove results on equi-attraction and residual continuity in a moredirect way, which also serves to demonstrate more clearly why these results aretrue. Given equi-attraction the attractor is the uniform limit of a sequence ofcontinuous functions, and hence continuous (the converse requires a generalisedversion of Dini’s Theorem); more generally, it is the pointwise limit of a sequence ofcontinuous functions, i.e. a ‘Baire one’ function, and therefore the set of continuitypoints forms a residual set.