Homeomorphism between one sided sequences and two sided sequences in shift level

Question

Let $\Sigma_{12}$ denote the set of one-sided sequences of $0$’s and $1$’s. Define $\phi \colon \Sigma_{12} \to \Sigma_{2}$ by $\phi(s_0s_1s_2\cdots) = (\cdots s_5s_3s_1 \cdot s_0s_2s_4 \cdots)$. How I can prove that $\phi$ is a homeomorphism. Any hint will be appreciated. Thank you.

Answer 1

Since you only ask for a hint:

First show that the map $\phi$ is a bijection. Then show that $\phi$ is continuous by considering the preimages of the open sets $\mathcal{Z}_{u}=\{s \in \Sigma_{2} \mid s = \cdots s_{-(n+2)}s_{-(n+1)}u s_{n} s_{n+1} \cdots\}$, where $u$ is a word of length $2n$ (these sets form a basis of $\Sigma_{2}$). That is, $\mathcal{Z}_u$ is the set of all infinite strings that have the word $u$ at the origin. Then use the fact that continuous bijections from compact spaces to Hausdorff spaces are homeomorphisms.