That  looks much like a “universal set” for  – it not only has as members the powerset of , and all subsets of , it also has the powerset of that powerset and so on – its members are closed under the operations of taking powerset or taking a subset.
That resembles a "universal set" – it encompasses not only the powerset of , but also all subsets of , as well as the powersets of those subsets, and so on – its members are closed under the operations of taking powersets or taking subsets.