Essential $P$-spaces: a generalization of door spaces

Osba, Emad Abu, and Henriksen, Melvin. "Essential $P$-spaces: a generalization of door spaces." Commentationes Mathematicae Universitatis Carolinae 45.3 (2004): 509-518. <http://eudml.org/doc/249348>.

@article{Osba2004,
abstract = {An element $f$ of a commutative ring $A$ with identity element is called a 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 $P$-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 $P$-space. If all but at most one point of $X$ is a $P$-point, then $X$ is called an 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.},
author = {Osba, Emad Abu, Henriksen, Melvin},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {$P$-point; $P$-space; essential $P$-space; door space; $F$-space; basically disconnected space; space of minimal prime ideals; $SV$-ring; $SV$-space; rank; von Neumann regular ring; von Neumann local ring; Lindelöf space; essential -space; -point; Lindelöf space; -point},
language = {eng},
number = {3},
pages = {509-518},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Essential $P$-spaces: a generalization of door spaces},
url = {http://eudml.org/doc/249348},
volume = {45},
year = {2004},
}

TY - JOUR
AU - Osba, Emad Abu
AU - Henriksen, Melvin
TI - Essential $P$-spaces: a generalization of door spaces
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2004
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 45
IS - 3
SP - 509
EP - 518
AB - An element $f$ of a commutative ring $A$ with identity element is called a 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 $P$-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 $P$-space. If all but at most one point of $X$ is a $P$-point, then $X$ is called an 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.
LA - eng
KW - $P$-point; $P$-space; essential $P$-space; door space; $F$-space; basically disconnected space; space of minimal prime ideals; $SV$-ring; $SV$-space; rank; von Neumann regular ring; von Neumann local ring; Lindelöf space; essential -space; -point; Lindelöf space; -point
UR - http://eudml.org/doc/249348
ER -