P-points in N∗ and the spaces of continuous functions

It is proved that a point φ from the Čech-Stone remainder N∗ is a P-point iff Cp(Nφ) is a hereditary Baire space, where Nφ = N ∪ {φ}. Some characterizations of P-points in terms of games played in Nφ and Cp(Nφ) are also given.