Let $A,B\subseteq\mathbb{N}$ and $d(A,B)=\sum_{n\in A\Delta B}2^{-n}$, where $A\Delta B = (A\cup B)\setminus (A\cap B)$, be a metric on subsets of the natural numbers. I'm asked to show that for Gale-Stewart games there does not exists a continuous function which decides the winner (when $d$ is used on subsets of $\mathbb{N}$). I've been hinted that, under $d$, we have $\lim_{n\rightarrow \infty} G_n=\emptyset$, where $G_n$ is the positions after $2n$ moves which guarantees player $I$ to win (which i suppose is contradictory and somehow this should lead to the conclusion that a continuous function deciding the winner does not exist).
Don't know where to start. Help.