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.
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, Lindel\"of space
AMS Subject Classification: 54D, 54G, 13F30, 16A30