My aim in this post (if I have enough space) is to prove that every closed game can be lifted to an open (and therefore, by continuity, which is part of the definition of lifting, clopen) game. Since I am discussing a formal proof, I shall be a little more formal with my definitions than I have been so far. Much of what I write to begin with will be repeating things I have already said in the two previous posts.
Trees and paths
Recall the definition of a pruned tree. This is an infinite rooted tree such that from every vertex there is at least one directed infinite path. (Less formally, if you are walking away from the root, you never get stuck.) Given such a tree , we write for the set of infinite directed paths in . If we are working in , then the tree we will work with has finite sequences as its vertices, with each sequence joined to its extensions . Then .