Inhabitants of interesting subsets of the Bousfield lattice

In 1979, Bousfield defined an equivalence relation on the stable homotopy category. The set of Bousfield classes has some important subsets such as the distributive lattice DL of all classes $〈E〉$ which are smash idempotent and the complete Boolean algebra cBA of closed classes. We provide examples of spectra that are in DL, but not in cBA; in particular, for every prime p, the Bousfield class of the Eilenberg–MacLane spectrum $〈H{\mathbb{F}}_p〉$ is in $\mathbf{DL}?\mathbf{cBA}$.