Stegall compact spaces which are not fragmentable

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 S. Namely the following is proved. (1) If aleph(1) is less than the least inaccessible cardinal in L and MA & inverted left perpendicular CH hold then there is a nonfragmentable compact Hausdorff space K such that every minimal usco mapping of a Baire space into K is singlevalued 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 singlevalued at points of a residual set. (C) 1999 Elsevier Science B.V. All rights reserved.