In Telgarsky - Topological games, in page 246, the following game $G(K,X)$ is described: There are given a space $X$ and a class $K$ of spaces such that $Y \in K \Rightarrow \mathcal F(Y) \subset K$. (Recall that $\mathcal F(Y)$ denotes the collection of all closed subsets of a space Y). Player ONE chooses an $H_0 \in \mathcal F(X) \cap K$, player TWO chooses an $E_1 \in \mathcal F(X)$, with $E_1 \subset X \setminus H_0$, Player ONE chooses an $H_1 \in \mathcal F(X) \cap K$, with $H_1 \subset E_1$, player TWO chooses an $E_2 \in \mathcal F(X)$, with $E_2 \subset E_1 \setminus H_1$, and so forth. Put $E_0 = X$. Player ONE wins the play $(E_0,H_0,E_1,H_1,...)$ if and only if $\bigcap_{n<\omega} E_n = \emptyset$.
If I understand correctly, the sets $E_i$, are closed subsets of $X$. So, isn't there allways at least one point in $\bigcap_{n<\omega} E_n$?
Thank you!