Topological games defined by ultrafilters

By using a free ultrafilter p on ω, we introduce an infinite game, called Gp(x,X)-game, played around a point x in a space X. This game is the natural generalization of the G(x,X)-game introduced by A. Bouziad. We establish some relationships between the Gp(x,X)-game and the Rudin–Keisler pre-order on ω∗. We prove that if p,q∈ω∗, then βω⧹PRK(p) is a Gq-space if and only if q≰RKp; and, for every p∈ω∗, there is a Gp-space that is not a Gq-space for every q∈T(p)⧹R(p), where R(p)={f̂(p): ∃A∈p(f|Ais strictly increasing)}. As a consequence, we characterize the Q-points in ω∗ as follows: p∈ω∗ is a Q-point iff every Gp-space is a Gq-space for every q∈T(p), where T(p)={q∈ω∗: p⩽RKq and q⩽RKp}.