Definition

A metric is a distance function d:X×X→[0,∞)d : X \times X \to [0,\infty) for set XX satisfying axioms

  • triangle inequality, d(x,y)+d(y,z)≥d(x,z)d(x,y) + d(y,z) \geq d(x,z)
  • point inequality, 0≥d(x,x)0 \geq d(x,x)
    • so 0=d(x,x)0 = d(x,x)
  • separation, x=yx=y if d(x,y)=0d(x,y) = 0
    • so x=yx=y iff d(x,y)=0d(x,y)=0
  • symmetry d(x,y)=d(y,x)d(x,y) = d(y,x)

References

  1. https://ncatlab.org/nlab/show/metric+space