Answer to Question #174700 in Functional Analysis for AR Ahmes

Question

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

Answer 1

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 $d(a,a)=0$.