Stegall compact spaces which are not fragmentable

Abstract

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 $\text{S}$. Namely the following is proved.

(1) If $\text{ℵ}{}_1$ is less than the least inaccessible cardinal in $\text{L}$ and $\text{MA}\text{\&}\text{¬}\text{CH}$ hold then there is a nonfragmentable compact Hausdorff space $\text{K}$ such that every minimal usco mapping of a Baire space into $\text{K}$ is singlevalued at points of a residual set.

(2) If $\text{V=L}$ then there is a nonfragmentable compact Hausdorff space $\text{K}$ such that every minimal usco mapping of a completely regular Baire space into $\text{K}$ is singlevalued at points of a residual set.