Consider the infinite cyclic group ℤ, represent its elements as the integer points on the real line, and define the distance between two elements as the distance between the corresponding points. It thus becomes a metric space, though not a very interesting one; all distances are integers, and the topology is discrete. Another possibility is to represent ℤ by all multiples of the number 1/2. Though the distances become smaller, they still comprise a discrete set, and the topology remains discrete. The same happens if we represent ℤ by the multiples of 1/3, or of 1/4, etc. But if we look at all of these representations, we have marked on the real line all rational points, which are a dense subset of the line. It makes sense to consider the full line as a sort of limit of these representations. Similarly, we can represent the free abelian group ℤ × ℤ by the so-called lattice points, i.e. all the points in the plane with integer coordinates, or by all points whose coordinates are integral multiples of 1/2, etc., and consider the full plane as a limit of these representations. A slightly different view-point is to consider the integer points on the line as the Cayley graph of ℤ. Then the distance is identical with the distance on the graph. Then we take the same graph, but divide the distances by 2, 3, etc. We can apply a similar process to any group.