Geodesic
Isomorphism Problems for the Baire Function Spaces of Topological Spaces
Serdica Mathematical Journal, Tome 24 (1998) no. 1, pp. 5-20.

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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(bα X), where bα X is a compactification of the space P X – 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.
Keywords: Baire Complemented Banach Space, Baire Function, Scattered Space, Baire Topology, D-Set 
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Choban, Mitrofan. Isomorphism Problems for the Baire Function Spaces of Topological Spaces. Serdica Mathematical Journal, Tome 24 (1998) no. 1, pp. 5-20. http://geodesic.mathdoc.fr/item/SMJ2_1998_24_1_a2/