This node is part of the game proof of the Baire category theorem, and won't make any sense unless you're coming from there!
Suppose G's complement is a dense open set. Alice goes first, and plays some finite sequence of bits; as in GpBCT: proof that Alice wins on an open set, this restricts the remainder of the game to some dyadic interval I; we may look at the remainder of the game as the game on the intersection G∩I (suitably moved and re-scaled), with Bob moving first. Since the intersection of G and I is non-empty (G is dense!) and open (we can ignore the edges, or say that it contains an open set, and use the remarks in the opening node), this is exactly the case of proving that a non-empty open set is a win for the first player.