# topology on n-colorings of a lattice?

Consider the collection of all $n$-colorings of $\mathbb{Z^{d}}$ (i.e. the collection of all ways to color each lattice point one of $n$ colors). What are some non-trivial ways to define a topology on this collection?

-

Hint: what topologies do you know on $\{0,1\}^{\mathbb{Z}}$, which is the set of $2$-colorings of $\mathbb{Z}^1$? Can you extend them to this case?