Space of Baire functions. I

Jayne, J. E.

Annales de l'Institut Fourier, Tome 24 (1974) no. 4, pp. 47-76.

On donne quelques conditions pour l’existence de fonctions réelles de Baire de toutes les classes sur certains espaces $\mathbf{K}$-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 $\Omega$ (= 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 $\alpha$ pour tous les ordinaux dénombrables $\alpha$. 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.

Several equivalent conditions are given for the existence of real-valued Baire functions of all classes on a type of $\mathbf{K}$-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 $\Omega$ (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 $\alpha$ for each countable ordinal $\alpha$. 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.

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