Let $A\subseteq\mathcal{P}(\mathbb{N})$ be any subset of naturals, the charasteristic function of A is for each $i\in\mathbb{N}$
$$\chi_{A}(i) = \begin{cases} 0 & i \notin A \\ 1 & i \in A \end{cases}$$
I would like to show that $\mathcal{P}(\mathbb{N})$ and the Cantor space $\{0,1\}^{\omega}$ is homeomorphic with the function for each $A\subseteq\mathcal{P}(\mathbb{N})$ $\phi(A)=\chi_{A}$. Surely it is bijection if we take any $f\in\{0,1\}^{\omega}$ and define $B=\{x\in\mathbb{N} : f(x)=1\} $ and $\chi_{B}=f$ but how can I show that $\phi$ and $\phi^{-1}$ is continuous ?