Full shift ($[k{]}^{\mathbb{Z}}$)

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 isAgreement 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.

On a two-sided full shift, the left shift is the homeomorphismIts inverse sends $z(j)$ to $z(j-1)$. On a one-sided sequence space, the same forward shift is generally not invertible.