Acta Mathematica Hungarica
Volume/Issue:
Volume 117: Issue 4
Metrizable compacta in the space of continuous functions with the topology of pointwise convergence
Author: V. Mykhaylyuk 1
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  • 1 Chernivtsi National University Department of Mathematical Analysis Kotsjubyns’koho 2 Chernivtsi 58012 Ukraine
DOI:
https://doi.org/10.1007/s10474-007-6105-7
Page Count:
315–323
Publication Date:
01 Dec 2007
Online Publication Date:
04 Oct 2007
Article Category:
Research Article
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Abstract  

We prove that every point-finite family of nonempty functionally open sets in a topological space X has the cardinality at most an infinite cardinal κ if and only if w(X) ≦ κ for every Valdvia compact space Y

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Cp(X). Correspondingly a Valdivia compact space Y has the weight at most an infinite cardinal κ if and only if every point-finite family of nonempty open sets in Cp(Y) has the cardinality at most κ, that is p(Cp(Y)) ≦ κ. Besides, it was proved that w(Y) = p(Cp(Y)) for every linearly ordered compact Y. In particular, a Valdivia compact space or linearly ordered compact space Y is metrizable if and only if p(Cp(Y)) = ℵ0. This gives answer to a question of O. Okunev and V. Tkachuk.

Cover Acta Mathematica Hungarica
Acta Mathematica Hungarica
Print ISSN:
0236-5294
Online ISSN:
1588-2632
Keywords:
Valdivia compact; metrizable compact; separately continuous function; point-finite cellularity; linearly ordered compact; 54E45; 54C35; 54A25; 54D30

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