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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?

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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?

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    The 'Hamming Distance' topology is discrete; e.g. the set B({1,0,0,0...}, 1.5) intersected with B({0,1,0,0,0...}, 1.5) intersected with B({0,0,1,0,0,0...}, 1.5) is just the one-point set {0,0,0,0,0...}.2011-12-31