Essential P-Spaces: A Generalization of Door Spaces

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 fg = f . A point p of a (Tychonoff) space X is called a P -point if each f in the ring C(X) of continuous realvalued 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.