Let $A$ be a non-empty finite set equipped with the discrete topology. Let $\mathbb{N}_0 := \{0,1,2,\dots\}$. Define $\Sigma_A := A^{\mathbb{Z}}$ and $\Sigma_A^+ := A^{\mathbb{N}_0}$, each equipped with its [product topology](/page/Product%20Topology). For $I \in \{\mathbb{Z},\mathbb{N}_0\}$, for every finite subset $F \subset I$, and for every function $a: F \to A$, define the finite-coordinate cylinder $C_I(F,a) := \{x \in A^I : x_i = a(i) \text{ for every } i \in F\}$. Then the collection of all sets $C_{\mathbb{Z}}(F,a)$, where $F \subset \mathbb{Z}$ is finite and $a:F\to A$, is a basis for the product topology on $\Sigma_A$, and the collection of all sets $C_{\mathbb{N}_0}(F,a)$, where $F \subset \mathbb{N}_0$ is finite and $a:F\to A$, is a basis for the product topology on $\Sigma_A^+$. Moreover, both $\Sigma_A$ and $\Sigma_A^+$ are compact metrizable spaces. The one-sided shift map $\sigma_+: \Sigma_A^+ \to \Sigma_A^+$ defined by $(\sigma_+(x))_i = x_{i+1}$ for every $x\in\Sigma_A^+$ and every $i\in\mathbb{N}_0$ is continuous. The two-sided shift map $\sigma: \Sigma_A \to \Sigma_A$ defined by $(\sigma(x))_i = x_{i+1}$ for every $x\in\Sigma_A$ and every $i\in\mathbb{Z}$ is a homeomorphism.