Prove that $Σ_m$ is homeomorphic to $Σ^+_m$, $m ≥ 2$

Question

Prove that $Σ_m$ is homeomorphic to $Σ^+_m$, $m ≥ 2$ where $Σ^+_m=\{0,1,...,m-1\}^{\Bbb N}$ : Set of all one sided sequences with entries in $\{0,1,...,m-1\}$ e.g $0.x_0x_1...$ where $x_i \in \{0,1,...,m-1\}$ and $Σ_m=\{0,1,...,m-1\}^{\Bbb Z}$ Set of all two sided sequences with entries in $\{0,1,...,m-1\}$ e.g $...x_{-2}x_{-1}.x_0x_1...$ where $x_i \in \{0,1,...,m-1\}$

It has a topology with the subbasis consisting of cylinders $\{C_j^n| n\in \Bbb N \cup 0$ or $\Bbb Z, j \in \{0,..,m-1\}\}$ where $C_j^n=\{x \in Σ_m$ or $Σ^+_m | x_n=j\}$

And it has a metric space structure for $x=(x_n)$ and $y=(y_n) \in Σ_m $ or $Σ^+_m$ define $d(x,y)=\frac1{2^L}$; where $L=\{|i|: x_i \neq y_i, i \in \Bbb N \cup 0$ or $\Bbb Z\}$

How to solve it? I am not getting anything.

Answer 1

Define $f$ from $Σ_m=\{0,1,...,m-1\}^{\Bbb Z}$ to $Σ^+_m=\{0,1,...,m-1\}^{\Bbb N}$ such that $f$ sends $\cdots ,x_{-2},x_{-1}.x_0x,_1,\cdots$ to $0.x_1,x_{-1},x_2,x_{-2}\cdots$.