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Stegall compact spaces which are not fragmentable. (English) Zbl 0991.54030

Summary: Using modifications of the well-known construction of “double-arrow” space we give consistent examples of nonfragmentable compact Hausdorff spaces which belong to Stegall’s class \({\mathcal S}\). Namely the following is proved.
(1) If \(\aleph_1\) is less than the least inaccessible cardinal in \(L\) and \(\text{MA}\& \neg \text{CH}\) hold then there is a nonfragmentable compact Hausdorff space \(K\) such that every minimal usco mapping of a Baire space into \(K\) is single-valued at points of a residual set.
(2) If \(V=L\) then there is a nonfragmentable compact Hausdorff space \(K\) such that every minimal usco mapping of a completely regular Baire space into \(K\) is single-valued at points of a residual set.

MSC:

54D80 Special constructions of topological spaces (spaces of ultrafilters, etc.)
54A35 Consistency and independence results in general topology
54C60 Set-valued maps in general topology
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