A Gale-Stewart game $G(A)$ is played on a set $A\subseteq\mathbb N^\mathbb N$. In this game, players p0 and p1 alternately pick a natural number, forming a sequence $\alpha:=\alpha_0\alpha_1\alpha_2\ldots$ The goal of p0 is to form a sequence that is in $A$, p1's goal is to prevent this. A strategy for p0 then is a function that maps all finite sequences (words) of even length onto a natural number indicating p0's next move. Analogously, a strategy for p1 can be defined. A game $G(A)$ is called determined if there exists a strategy for either p0 or p1 that yields a certain win for him.
Now the Gale-Stewart theorem says that if $A$ is an open set, then $G(A)$ is determined. It seems logical (and is indeed true) that if $A$ is closed, then $G(A)$ is also determined, by considering a winning strategy $f$ in $G(\mathbb N^\mathbb N\setminus A)$ for p0, $f$ can be a winning strategy for p1 in $G(A)$. However, $f$ is a function from all words of even length to $\mathbb N$, and for it to be a strategy for p1, it must be a functions over words of odd length.
Now my question is, how can I easily see that if $A$ is closed, then also $G(A)$ is determined, as a corollary of the Gale-Stewart Theorem