Gillman and Henriksen also define a P-point as a point at which any prime ideal of the ring of real-valued continuous functions is maximal, and a P-space is a space in which every point is a P-point.
[2] Different authors restrict their attention to topological spaces that satisfy various separation axioms.
With the right axioms, one may characterize P-spaces in terms of their rings of continuous real-valued functions.
A different notion of a P-space has been introduced by Kiiti Morita in 1964, in connection with his (now solved) conjectures (see the relevant entry for more information).
A notion of a p-space has been introduced by Alexander Arhangelskii.