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 f2g=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 η is a Gδ, and not even then if there are infinitely many pairwise disjoint cozerosets with η in their closure. The general case is considered and open problems are posed.
© 2004 Charles University in Prague
Osba, Emad Abu, and Melvin Henriksen. "Essential P-spaces: a generalization of door spaces." Commentationes Mathematicae Universitatis Carolinae 45.3 (2004): 509-518.