# 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?

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## 1 Answer

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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Thanks....subspace topology of the box/product topology? Are there any other that you know of? I'm playing around with a function defined on this collection and would like to have different topologies to work with. –  user13255 Dec 1 '11 at 23:39
The discrete topology is another one unless it counts as trivial. If you define a "metric" in the spirit of Hamming distance (even though the value may be infinite) is that the same as the product topology? –  Ross Millikan Dec 1 '11 at 23:52
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...}. –  Lopsy Dec 31 '11 at 23:48