Orihuela, José On the semiconvex category of hyperplanes which are products of topological Baire vector spaces. (Spanish. English summary) Zbl 0701.46001 Rev. R. Acad. Cienc. Exactas Fís. Nat. Madr. 82, No. 3-4, 425-438 (1988). Summary: We prove in this paper that any dense hyperplane in a topological vector Baire space or any topological product of a family of topological vector Baire spaces cannot be covered by a sequence of closed semiconvex subsets without an interior point. In such a way, extensions of the known results for convex subsets of M. Valdivia [Topics in locally convex spaces (1982; Zbl 0489.46001)] are obtained. We reach this results dealing with the class of Baire semiconvex spaces which is related to the Baire-convex spaces of M. Valdivia. Proving their stability properties we obtain our conclusions. We give examples, even in the category of normed spaces, for distinguishing between Baire, Baire semi-convex and Baire-convex spaces. Our results are related to the negative answers to the Wilansky-Klee conjecture, asking if every dense hyperplane of a Banach space must be of second category, and to the general problem of stability of Baire spaces by topological products. MSC: 46A08 Barrelled spaces, bornological spaces Keywords:dense hyperplane in a topological vector Baire space; topological product of a family of topological vector Baire spaces; Baire semiconvex spaces; stability properties; negative answers to the Wilansky-Klee conjecture Citations:Zbl 0489.46001 PDFBibTeX XMLCite \textit{J. Orihuela}, Rev. R. Acad. Cienc. Exactas Fís. Nat. Madr. 82, No. 3--4, 425--438 (1988; Zbl 0701.46001) Full Text: EuDML