I am studying about topological game, and I found the progression:
Point-Open(X)>> Finite-Open(X)>>Compact-Open(X).
I understand that a natural way to extend 'be finite' is 'be compact', but looking from a cardinal point of view, the sequence would be 1 (Point) >> n (Finite)>> $\omega$ (Countable).
I have not got a book that at least mentions the possibility of the countable-open game, for this reason I would like to ask if you know any book that mentions it, or books that develop some of the other games mentioned. My goal is to see if there is any relationship between countable-open game and the others already known.