ISOMORPHISM PROBLEMS FOR THE BAIRE FUNCTION SPACES OF TOPOLOGICAL SPACES

Dedicated to the memory of Professor D. Doitchinov Abstract. Let a compact Hausdorff space X contain a non-empty perfect subset. If α < β and β is a countable ordinal, then the Banach space Bα(X) of all bounded real-valued functions of Baire class α on X is a proper subspace of the Banach space Bβ(X). In this paper it is shown that: 1. Bα(X) has a representation as C(bαX), where bαX is a compacti-fication of the space PX – the underlying set of X in the Baire topology generated by the Gδ-sets in X. 2. If 1 ≤ α < β ≤ Ω, where Ω is the first uncountable ordinal number, then Bα(X) is uncomplemented as a closed subspace of Bβ(X). These assertions for X = [0, 1] were proved by W. G. Bade [4] and in the case when X contains an uncountable compact metrizable space – by F.K.Dashiell [9]. Our argumentation is one non-metrizable modification of both Bade’s and Dashiell’s methods.