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A nonnegative, real-valued function, LaTeX: d, on pairs from a set LaTeX: S such that for each LaTeX: x, LaTeX: y, and LaTeX: z in LaTeX: S:

  1. LaTeX: d(x, y) \ge 0;
  2. LaTeX: d(x, y) = 0 \Rightarrow x=y;
  3. LaTeX: d(x, y) + d(y, z) \ge d(x, z).

The function is also called a distance; the pair LaTeX: (S, d) is a metric space. Condition (3) is called the Triangle inequality. A space is metrizable if a metric can be defined. A metric is induced by a norm if one exists: LaTeX: d(x,y) = \| x - y \|. See Hausdorff metric.

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