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Title: The $\sigma$-property in $C(X)$ (English)
Author: Hager, Anthony W.
Language: English
Journal: Commentationes Mathematicae Universitatis Carolinae
ISSN: 0010-2628 (print)
ISSN: 1213-7243 (online)
Volume: 57
Issue: 2
Year: 2016
Pages: 231-239
Summary lang: English
Category: math
Summary: The $\sigma$-property of a Riesz space (real vector lattice) $B$ is: For each sequence $\{b_{n}\}$ of positive elements of $B$, there is a sequence $\{\lambda_{n}\}$ of positive reals, and $b\in B$, with $\lambda_{n}b_{n}\leq b$ for each $n$. This condition is involved in studies in Riesz spaces of abstract Egoroff-type theorems, and of the countable lifting property. Here, we examine when ``$\sigma$'' obtains for a Riesz space of continuous real-valued functions $C(X)$. A basic result is: For discrete $X$, $C(X)$ has $\sigma$ iff the cardinal $|X|< \mathfrak{b}$, Rothberger's bounding number. Consequences and generalizations use the Lindelöf number $L(X)$: For a $P$-space $X$, if $L(X)\leq \mathfrak{b}$, then $C(X)$ has $\sigma$. For paracompact $X$, if $C(X)$ has $\sigma$, then $L(X)\leq \mathfrak{b}$, and conversely if $X$ is also locally compact. For metrizable $X$, if $C(X)$ has $\sigma$, then $X$ \textit{is} locally compact. (English)
Keyword: Riesz space
Keyword: $\sigma$-property
Keyword: bounding number
Keyword: $P$-space
Keyword: paracompact
Keyword: locally compact
MSC: 03E17
MSC: 06F20
MSC: 46A40
MSC: 54A25
MSC: 54C30
MSC: 54D20
MSC: 54D45
MSC: 54G10
idZBL: Zbl 06604503
idMR: MR3513446
DOI: 10.14712/1213-7243.2015.162
Date available: 2016-07-05T15:10:41Z
Last updated: 2018-07-02
Stable URL:
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