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  7. Isomorphism Problems for the Baire Function Spaces of Topological Spaces

Isomorphism Problems for the Baire Function Spaces of Topological Spaces

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Authors: Choban, Mitrofan
Issue Date: 1998
Citation: Serdica Mathematical Journal, Vol. 24, No 1, (1998), 5p-20p Copy to clipboard
ISSN: 1310-6600
URI: http://hdl.handle.net/10525/542 Copy to clipboard
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(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.
Language: en
Publisher: Institute of Mathematics and Informatics, Bulgarian Academy of SciencesSubject: Baire Complemented Banach SpaceBaire FunctionScattered SpaceBaire TopologyD-Set
Type: Article