$P$-points in random universes

A point in a topological space is called a P-point if the intersection of any countable family of its neighborhood is a neighborhood. Shelah [8] has recently shown the existence of a P-point in f3N \ N to be independent of ZFC. Various assumptions are known to imply the existence of P-points [1], [4], [6]. In this paper we contribute a new axiom of this sort and we show that the axiom holds in any random extension of a model of ZFC + CH. The referee has pointed out that Kunen has shown the existence of P-points in certain models of this description [5].