# Isomorphism Problems for the Baire Function Spaces of Topological Spaces

Serdica Mathematical Journal (1998)

- Volume: 24, Issue: 1, page 5-20
- ISSN: 1310-6600

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topChoban, 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 -

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