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Title: Order intervals in $C(K)$. Compactness, coincidence of topologies, metrizability (English)
Author: Lipecki, Zbigniew
Language: English
Journal: Commentationes Mathematicae Universitatis Carolinae
ISSN: 0010-2628 (print)
ISSN: 1213-7243 (online)
Volume: 63
Issue: 3
Year: 2022
Pages: 295-306
Summary lang: English
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Category: math
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Summary: Let $K$ be a compact space and let $C(K)$ be the Banach lattice of real-valued continuous functions on $K$. We establish eleven conditions equivalent to the strong compactness of the order interval $[0,x]$ in $C(K)$, including the following ones: (i) $\{x>0\}$ consists of isolated points of $K$; (ii) $[0,x]$ is pointwise compact; (iii) $[0,x]$ is weakly compact; (iv) the strong topology and that of pointwise convergence coincide on $[0,x]$; (v) the strong and weak topologies coincide on $[0,x]$. \noindent Moreover, the weak topology and that of pointwise convergence coincide on $[0,x]$ if and only if $\{x>0\}$ is scattered. Finally, the weak topology on $[0,x]$ is metrizable if and only if the topology of pointwise convergence on $[0,x]$ is such if and only if $\{x>0\}$ is countable. (English)
Keyword: real linear lattice
Keyword: order interval
Keyword: locally solid
Keyword: Banach lattice $C(K)$
Keyword: strongly compact
Keyword: weakly compact
Keyword: pointwise compact
Keyword: coincidence of topologies
Keyword: metrizable
Keyword: scattered
Keyword: Čech--Stone compactification
MSC: 46A40
MSC: 46B42
MSC: 46E05
MSC: 54C35
MSC: 54D30
idZBL: Zbl 07655801
idMR: MR4542790
DOI: 10.14712/1213-7243.2022.006
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Date available: 2023-02-01T12:03:11Z
Last updated: 2024-10-04
Stable URL: http://hdl.handle.net/10338.dmlcz/151477
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