no. 4
Space of Baire functions. I
Annales de l'Institut Fourier, Tome 24 (1974) no. 4, pp. 47-76

Several equivalent conditions are given for the existence of real-valued Baire functions of all classes on a type of K-analytic spaces, called disjoint analytic spaces, and on all pseudocompact spaces. The sequential stability index for the Banach space of bounded continuous real-valued functions on these spaces is shown to be either 0,1, or Ω (the first uncountable ordinal). In contrast, the space of bounded real-valued Baire functions of class 1 is shown to contain closed linear subspaces with index α for each countable ordinal α. The sequential stability index for linear subspaces of continuous real-valued functions on a compact space is shown to be invariant under isomorphic embeddings in the space of continuous real-valued functions on any compact space.

On donne quelques conditions pour l’existence de fonctions réelles de Baire de toutes les classes sur certains espaces K-analytiques (appelés espaces analytiques disjoints) et sur tous les espaces pseudo-compacts. On montre que l’indice de stabilité séquentielle de l’espace de Banach des fonctions réelles bornées et continues est égal à 0,1 ou Ω (= premier ordinal non dénombrable) sur ces espaces. Au contraire, on montre que l’espace des fonctions de Baire réelles bornées de la première classe contient des sous-espaces linéaires fermés de l’indice α pour tous les ordinaux dénombrables α. On montre que l’indice de stabilité séquentielle des sous-espaces linéaires des fonctions réelles continues sur un compact reste invariant par rapport à l’immersion isomorphique dans l’espace des fonctions réelles continues sur un compact quelconque.

Jayne, J. E. Space of Baire functions. I. Annales de l'Institut Fourier, Tome 24 (1974) no. 4, pp. 47-76. doi: 10.5802/aif.531
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