Essential P-Spaces: A Generalization of Door Spaces
Essential P-Spaces: A Generalization of Door Spaces
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发表时间：
2004
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通讯作者：
E. A. Osba;M. Henriksen
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作者：
E. A. Osba;M. Henriksen
一个有单位元的交换环A的元素f称为冯·诺依曼正则元，如果在A中有g使得f^{2}g=f。（Tychonoff）空间$X$的点$p$称为$P$-{\it_point\/}，如果连续实值函数环$C（X）$中的每个$f$在$p$的邻域上是常数。众所周知，环$C（X）$是冯诺依曼正则环当且仅当它的每个元素都是冯诺依曼正则元素;在这种情况下，$X$被称为$P$-{\it\/}。如果$X$中除了至多一个点之外的所有点都是$P$-点，则$X$被称为{\it essential $P$-空间\/}。在早期的工作中，我们证明了X是本质P-空间当且仅当对于C（X）中的每一个f，f或1-f都是冯诺依曼正则元。本质$P$-空间（它是J.L. Kelley的门空间）的代数性质的帮助下得出的$C（X）$。尽管本质P-空间的描述听起来很简单，但它并不简单，除非它的非P-点$\eta$是G_{\delta}$，甚至如果有无穷多个两两不相交的余零集在它们的闭包中有$\eta$，它也不简单。一般情况下被认为是开放的问题。
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.