| 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:
|
http://hdl.handle.net/10338.dmlcz/145753 |
| . |
| Reference:
|
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| Reference:
|
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| Reference:
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| . |
