Isomorphism Problems for the Baire Function Spaces of Topological Spaces

Choban, Mitrofan. "Isomorphism Problems for the Baire Function Spaces of Topological Spaces." Serdica Mathematical Journal 24.1 (1998): 5-20. <http://eudml.org/doc/11574>.

@article{Choban1998,
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.},
author = {Choban, Mitrofan},
journal = {Serdica Mathematical Journal},
keywords = {Baire Complemented Banach Space; Baire Function; Scattered Space; Baire Topology; D-Set; Baire complemented Banach space; Baire function; Baire topology; scattered space; -set; Stone-Čech compactification},
language = {eng},
number = {1},
pages = {5-20},
publisher = {Institute of Mathematics and Informatics Bulgarian Academy of Sciences},
title = {Isomorphism Problems for the Baire Function Spaces of Topological Spaces},
url = {http://eudml.org/doc/11574},
volume = {24},
year = {1998},
}

TY - JOUR
AU - Choban, Mitrofan
TI - Isomorphism Problems for the Baire Function Spaces of Topological Spaces
JO - Serdica Mathematical Journal
PY - 1998
PB - Institute of Mathematics and Informatics Bulgarian Academy of Sciences
VL - 24
IS - 1
SP - 5
EP - 20
AB - 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.
LA - eng
KW - Baire Complemented Banach Space; Baire Function; Scattered Space; Baire Topology; D-Set; Baire complemented Banach space; Baire function; Baire topology; scattered space; -set; Stone-Čech compactification
UR - http://eudml.org/doc/11574
ER -