Type: Article

Topological games defined by ultrafilters

S. García-Ferreira; R. A. González-Silva;

Abstract:

By using a free ultrafilter p on omega, we introduce an infinite game, called G(p)(x, X)-game, played around a point x in a space X. This game is the natural generalization of the 9(x, X)-game introduced by A. Bouziad. We establish some relationships between the G(p)(x, X)-game and the Rudin-Keisler pre-order on omega*. We prove that if p, q is an element of omega*, then betaomega \ P(RK) (P) is a G(p)-space if and only if q less than or equal to (RK) p; and, for every p is an element of omega*, there is a G(p)-space that is not a G(q)-space for every q is an element of T(p) \ R(p), where R(p) = {(f) over cap (p): There ExistsA is an element of p(f\(A)is strictly increasing)}. As a consequence, we characterize the Q-points in omega* as follows: p is an element of omega* is a Q-point iff every G(p)-space is a G(q)-space for every q is an element of T(p), where T(p) = {q is an element of omega*: P less than or equal to(RK) q and q less than or equal to(RK) P}. (C) 2003 Elsevier B.V. All rights reserved.