The ηx-sets of Hausdorff have large compactifications (of cardinality ≽ exp(α); and of cardinality ≽ exp(exp(2\u3cα)) in the Stone-Čech case). If Qα denotes the unique (when it exists) ηα -set of cardinality α, then Qα can be decomposed (= partitioned) into homeomorphs of any prescribed nonempty subspace; moreover the subspaces of Qα can be characterized as those which arc regular T1, of cardinality and weight ≼ α, whose topologies are closed under \u3c α intersections
