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Polish Academy of Sciences, Institute of Mathematics
The ηx-sets of Hausdorff have large compactifications (of cardinality ≽ exp(α); and of cardinality ≽ exp(exp(2<α)) 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 < α intersections.
Bankston, Paul, "Topological Extensions and Subspaces of ηα-sets" (1983). Mathematics, Statistics and Computer Science Faculty Research and Publications. 201.