Question #174700

Find all metrics on a set X consisting of two points. Consisting of one point only.


1
Expert's answer
2021-03-26T03:57:30-0400

Suppose first that X={a,b}X=\{a,b\}. A metric dd on XX is a positive-valued function on X×X={(a,a),(b,b),(a,b),(b,a)}X\times X= \{(a,a), (b,b), (a,b),(b,a) \} :

  • We should have d(a,a)=d(b,b)=0d(a,a)=d(b,b)=0
  • We should have d(a,b)=d(b,a)>0d(a,b)=d(b,a)>0. Let us call dλd_\lambda a function such that dλ(a,b)=λ>0d_\lambda (a,b)=\lambda>0

This family of functions describes all possible metrics on a two-point space, as we can verify that for all λ>0\lambda>0 the function dλd_\lambda satisfies the triangle inequality trivially and thus they are metrics.

If XX is a one-point space, there is a unique metric on X×X={(a,a)}X\times X=\{ (a,a)\} that gives d(a,a)=0d(a,a)=0.


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