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The smallest basically disconnected preimage of a space. (English) Zbl 0593.54036
The author shows that for every completely regular Hausdorff space X there exists the smallest basically disconnected space Λ X which has a canonical perfect irreducible mapping onto X; i.e. there exists a perfect irreducible mapping Λ : Λ X onto X such that for every perfect irreducible mapping g:Y onto X, where Y is basically disconnected, there exists a continuous mapping h:Y onto ΛX such that g=Λh. In the first stage of the construction the author proves that the space Λ 1 X consisting of all prime prime-z-filters which are generated by open ultrafilters is homeomorphic to X iff X is basically disconnected. Next the space Λ X is constructed as an inverse limit of a continuous inverse sequence {Λ α X,Λ β α ; β<α<ω 1 }, where Λ α+1 X=Λ 1 (Λ α X) for every α<ω 1 . From the existence of the space Λ X it follows that for every locally compact basically disconnected space X there exists the smallest basically disconnected compactification BX.
Reviewer: A.Błaszczyk

MSC:
54G05Extremally disconnected spaces, F-spaces, etc.
54D35Extensions of topological spaces (compactifications, supercompactifications, completions, etc.)
54C10Special maps on topological spaces