| Title:
|
Spaces $X$ in which all prime $z$-ideals of $C(X)$ are minimal or maximal (English) |
| Author:
|
Henriksen, Melvin |
| Author:
|
Martínez, Jorge |
| Author:
|
Woods, R. Grant |
| Language:
|
English |
| Journal:
|
Commentationes Mathematicae Universitatis Carolinae |
| ISSN:
|
0010-2628 (print) |
| ISSN:
|
1213-7243 (online) |
| Volume:
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44 |
| Issue:
|
2 |
| Year:
|
2003 |
| Pages:
|
261-294 |
| . |
| Category:
|
math |
| . |
| Summary:
|
Quasi $P$-spaces are defined to be those Tychonoff spaces $X$ such that each prime $z$-ideal of $C(X)$ is either minimal or maximal. This article is devoted to a systematic study of these spaces, which are an obvious generalization of $P$-spaces. The compact quasi $P$-spaces are characterized as the compact spaces which are scattered and of Cantor-Bendixson index no greater than 2. A thorough account of locally compact quasi $P$-spaces is given. If $X$ is a cozero-complemented space and every nowhere dense zeroset is a $z$-embedded $P$-space, then $X$ is a quasi $P$-space. Conversely, if $X$ is a quasi $P$-space and $F$ is a nowhere dense $z$-embedded zeroset, then $F$ is a $P$-space. On the other hand, there are examples of countable quasi $P$-spaces with no $P$-points at all. If a product $X\times Y$ is normal and quasi $P$, then one of the factors must be a $P$-space. Conversely, if one of the factors is a compact quasi $P$-space and the other a $P$-space then the product is quasi $P$. If $X$ is normal and $X$ and $Y$ are cozero-complemented spaces and $f:X\longrightarrow Y$ is a closed continuous surjection which has the property that $f^{-1}(Z)$ is nowhere dense for each nowhere dense zeroset $Z$, then if $X$ is quasi $P$, so is $Y$. The converse fails even with more stringent assumptions on the map $f$. The paper then closes with a number of open questions, amongst which the most glaring is whether the free union of quasi $P$-spaces is always quasi $P$. (English) |
| Keyword:
|
quasi $P$-space |
| Keyword:
|
$P$-space |
| Keyword:
|
scattered space |
| Keyword:
|
Cantor-Bendixson derivatives |
| Keyword:
|
\newline nodec space |
| Keyword:
|
quasinormality |
| MSC:
|
06F25 |
| MSC:
|
54C10 |
| MSC:
|
54C40 |
| MSC:
|
54D45 |
| MSC:
|
54G10 |
| MSC:
|
54G12 |
| MSC:
|
54G99 |
| idZBL:
|
Zbl 1098.54013 |
| idMR:
|
MR2026163 |
| . |
| Date available:
|
2009-01-08T19:29:13Z |
| Last updated:
|
2020-02-20 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/119385 |
| . |
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