| Title:
|
On $a$-Kasch spaces (English) |
| Author:
|
Estaji, Ali Akbar |
| Author:
|
Henriksen, Melvin |
| Language:
|
English |
| Journal:
|
Archivum Mathematicum |
| ISSN:
|
0044-8753 (print) |
| ISSN:
|
1212-5059 (online) |
| Volume:
|
46 |
| Issue:
|
4 |
| Year:
|
2010 |
| Pages:
|
251-262 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
If $X$ is a Tychonoff space, $C(X)$ its ring of real-valued continuous functions. In this paper, we study non-essential ideals in $C(X)$. Let $a$ be a infinite cardinal, then $X$ is called $a$-Kasch (resp. $\bar{a}$-Kasch) space if given any ideal (resp. $z$-ideal) $I$ with $\operatorname{gen}\,(I)<a$ then $I$ is a non-essential ideal. We show that $X$ is an $\aleph _0$-Kasch space if and only if $X$ is an almost $P$-space and $X$ is an $\aleph _1$-Kasch space if and only if $X$ is a pseudocompact and almost $P$-space. Let $C_F(X)$ denote the socle of $C(X)$. For a topological space $X$ with only a finite number of isolated points, we show that $X$ is an $a$-Kasch space if and only if $\frac{C(X)}{C_F(X)}$ is an $a$-Kasch ring. (English) |
| Keyword:
|
$a$-Kasch space |
| Keyword:
|
almost $P$-space |
| Keyword:
|
basically disconnected |
| Keyword:
|
$C$-embedded |
| Keyword:
|
essential ideal |
| Keyword:
|
extremally disconnected |
| Keyword:
|
fixed ideal |
| Keyword:
|
free ideal |
| Keyword:
|
Kasch ring |
| Keyword:
|
$P$-space |
| Keyword:
|
pseudocompact space |
| Keyword:
|
Stone-Čech compactification |
| Keyword:
|
socle |
| Keyword:
|
realcompactification |
| MSC:
|
13A30 |
| MSC:
|
16S60 |
| MSC:
|
46J10 |
| MSC:
|
54C40 |
| idZBL:
|
Zbl 1240.54064 |
| idMR:
|
MR2754064 |
| . |
| Date available:
|
2010-12-14T14:56:55Z |
| Last updated:
|
2013-09-19 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/141380 |
| . |
| Reference:
|
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| Reference:
|
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| Reference:
|
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| Reference:
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| Reference:
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| Reference:
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| Reference:
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| Reference:
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| . |
