To recap briefly, we have defined a notion of lifting from one game to another and embarked on a proof that every Borel game can be lifted to a clopen game. The stages of that proof are to show that every closed game can be lifted to a clopen game, and that the set of games that can be lifted is closed under countable unions. It is straightforward to show that it is also closed under taking complements, so that will prove that all Borel games can be lifted to clopen games. As we saw earlier, if a game can be lifted to a determined game, then it is itself determined. Since clopen games are determined, this will be enough.
We are in the middle of the proof that closed games can be lifted to clopen games. We have defined the game to which we will lift, and shown how to map Player I strategies for the auxiliary game to strategies for the original game . So the first thing to do in this post is to show how to map Player II strategies for to Player II strategies for .