Space of Baire functions. I

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.