Example 40.1.18 (The Shift).label Let $\integer \curvearrowright 2^{\integer}$ be the Bernoulli shift of $\integer$, then:

1. (1)

   For any $x, y \in 2^{\integer}$, $y \in \ol{\integer x}$ if and only if for each $A \subset \integer$, there exists $n \in \integer$ such that $y|_{A} = nx|_{A}$. In other words, $x$ contains a copy of any segment of $y$^[1].

2. (2)

   $\integer \curvearrowright 2^{\integer}$ is point transitive, and hence topologically transitive^[2].

3. (3)