Topological games and some related topics

In this study, we generalize the Banach-Mazur, Choquet and point-open games. By using generalized point-open game, we find a characterization of compact scattered spaces. We get some relations between the other generalized games and directed complete spaces. There is an open question in the Choquet game about existence of NONEMPTY's 1-tactic, whenever s/he has a Markov strategy in this game (Galvin). We prove that if NONEMPTY has a Markov strategy in the Choquet game, then s\he has a 2-tactic in this game. This result is a partial answer for Galvin's question. More general version of this question is that if NONEMPTY has a k-Markov strategy in the Choquet game, does NONEMPTY have a k-tactic in this game? In some special topological spaces, we give some affirmative answers to this general question. For instance, we prove that if NONEMPTY has a k-Markov strategy in the Chouquet game on topological groups or on some spaces with some special bases, then s\he has a k-tactic in that game. We also show that if NONEMPTY has a winning strategy in the Choquet game on some spaces with some proper Noetherian bases, then NONEMPTY has a 1-tactic, in that game. We investigate some similar results for the Banach-Mazur game. We define some finite topological games and by using them we get some dimension functions which characterize the dimension function Ind and ind. By using one of the finite games, we get a game dimension function (ab). We prove that ab(X) = Ind(X) where X is a hereditarily normal space and Ind(X) does not exceed ab(X) where X is a normal space.