| Title:
|
Essential $P$-spaces: a generalization of door spaces (English) |
| Author:
|
Osba, Emad Abu |
| Author:
|
Henriksen, Melvin |
| Language:
|
English |
| Journal:
|
Commentationes Mathematicae Universitatis Carolinae |
| ISSN:
|
0010-2628 (print) |
| ISSN:
|
1213-7243 (online) |
| Volume:
|
45 |
| Issue:
|
3 |
| Year:
|
2004 |
| Pages:
|
509-518 |
| . |
| Category:
|
math |
| . |
| Summary:
|
An element $f$ of a commutative ring $A$ with identity element is called a {\it von Neumann regular element\/} if there is a $g$ in $A$ such that $f^{2}g=f$. A point $p$ of a (Tychonoff) space $X$ is called a $P$-{\it point\/} if each $f$ in the ring $C(X)$ of continuous real-valued functions is constant on a neighborhood of $p$. It is well-known that the ring $C(X)$ is von Neumann regular ring iff each of its elements is a von Neumann regular element; in which case $X$ is called a $P$-{\it space\/}. If all but at most one point of $X$ is a $P$-point, then $X$ is called an {\it essential $P$-space\/}. In earlier work it was shown that $X$ is an essential $P$-space iff for each $f$ in $C(X)$, either $f$ or $1-f$ is von Neumann regular element. Properties of essential $P$-spaces (which are generalizations of J.L. Kelley's door spaces) are derived with the help of the algebraic properties of $C(X)$. Despite its simple sounding description, an essential $P$-space is not simple to describe definitively unless its non $P$-point $\eta$ is a $G_{\delta}$, and not even then if there are infinitely many pairwise disjoint cozerosets with $\eta$ in their closure. The general case is considered and open problems are posed. (English) |
| Keyword:
|
$P$-point |
| Keyword:
|
$P$-space |
| Keyword:
|
essential $P$-space |
| Keyword:
|
door space |
| Keyword:
|
$F$-space |
| Keyword:
|
basically disconnected space |
| Keyword:
|
space of minimal prime ideals |
| Keyword:
|
$SV$-ring |
| Keyword:
|
$SV$-space |
| Keyword:
|
rank |
| Keyword:
|
von Neumann regular ring |
| Keyword:
|
von Neumann local ring |
| Keyword:
|
Lindelöf space |
| MSC:
|
13F30 |
| MSC:
|
16A30 |
| MSC:
|
16E50 |
| MSC:
|
54G10 |
| MSC:
|
54H13 |
| idZBL:
|
Zbl 1100.54024 |
| idMR:
|
MR2103145 |
| . |
| Date available:
|
2009-05-05T16:47:00Z |
| Last updated:
|
2012-04-30 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/119478 |
| . |
| Reference:
|
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