Academic Journal
The $\sigma$-property in $C(X)$
| العنوان: | The $\sigma$-property in $C(X)$ |
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| المؤلفون: | Hager, Anthony W. |
| بيانات النشر: | Charles University in Prague, Faculty of Mathematics and Physics |
| سنة النشر: | 2016 |
| المجموعة: | DML-CZ (Czech Digital Mathematics Library) |
| مصطلحات موضوعية: | 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 |
| الوصف: | 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. |
| نوع الوثيقة: | text |
| وصف الملف: | application/pdf |
| اللغة: | English |
| تدمد: | 0010-2628 1213-7243 |
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| الاتاحة: | http://hdl.handle.net/10338.dmlcz/145753 |
| Rights: | access:Unrestricted ; rights:DML-CZ Czech Digital Mathematics Library, http://dml.cz/ ; rights:Institute of Mathematics AS CR, http://www.math.cas.cz/ ; conditionOfUse:http://dml.cz/use |
| رقم الانضمام: | edsbas.17CA998 |
| قاعدة البيانات: | BASE |
| تدمد: | 00102628 12137243 |
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