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$P$-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
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.
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