Baire Complemented Banach Space Baire Function Scattered Space Baire Topology D-Set
Institute of Mathematics and Informatics Bulgarian Academy of Sciences
Serdica Mathematical Journal, Vol. 24, No 1, (1998), 5p-20p
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  and in
the case when X contains an uncountable compact metrizable space – by
F.K.Dashiell . Our argumentation is one non-metrizable modification of
both Bade’s and Dashiell’s methods.