metric space

In mathematics, a metric space consists of a set X, the “space,” for which the distance between all of it’s members can be calculated using a function d, the metric.

d must be a real function defined on X^2, and for each x,y,z \in X the following must all be true:

  • d(x,y) > 0 and d(x,y) = 0 \iff x = y (non-negative, identity of indiscernibles)
  • d(x,y) = d(y,x) (symmetry)
  • d(x,y) \leq d(x,z) + d(z,y) (triangle inequality)

d is also known as the distance function or just distance.

The most common metric space is 3-dimensional Euclidean space.

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