EMIS ELibM Electronic Journals Bulletin, Classe des Sciences Mathématiques et Naturelles, Sciences mathématiques naturelles / sciences mathematiques
Vol. CXXXI, No. 30, pp. 29–45 (2005)

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Representing trees as relatively compact subsets of the first Baire class

S. Todorcevic

Matematicki Institut, Kneza Mihaila 35, 11001 Beograd, Serbia and Montenegro, e-mail:stevo@mi.sanu.ac.yu

Abstract: We show that there is a scattered compact subset $K$ of the first Baire class, a Baire space $X$ and a separately continuous mapping $f:X\times K\arr{\mbb R}$ which is not continuous on any set of the form $G\times K$, where $G$ is a comeager subset of $X$. We also show that it is possible to have a scattered compact subset $K$ of the first Baire class which does have the Namioka property though its function space ${\mcal C}(K)$ fails to have an equivalent Frechet-differentiable norm and its weak topology fails to be $\sigma$-fragmented by the norm.

Keywords: Baire Class-1, Function spaces, Renorming

Classification (MSC2000): 46B03, 46B05

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Electronic fulltext finalized on: 21 Nov 2005. This page was last modified: 20 Jun 2011.

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