Answer to Question #174700 in Functional Analysis for AR Ahmes

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\\}". A metric "d" on "X" is a positive-valued function on "X\\times X= \\{(a,a), (b,b), (a,b),(b,a) \\}" :

  • We should have "d(a,a)=d(b,b)=0"
  • We should have "d(a,b)=d(b,a)>0". Let us call "d_\\lambda" a function such that "d_\\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 "\\lambda>0" the function "d_\\lambda" satisfies the triangle inequality trivially and thus they are metrics.

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


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