= Full shift
{title2=$[k]^{\mathbb Z}$}
The two-sided <full shift> on a finite <alphabet> $[k]$ consists of all <functions> $z:\mathbb Z\to[k]$, with the <product topology> and the <left shift>. It is a <compact metric space>. A <compatible metric> is
$$
d(z,w)=\sum_{j\in\mathbb Z}2^{-|j|-2}\mathbf1_{\{z(j)\ne w(j)\}}.
$$
Agreement on increasingly large finite coordinate sets is equivalent to convergence in this topology. The <metric> above is compatible but is not invariant under the <left shift>.
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