All metrics on set of size 2

Question

I was trying to determine all metrics possible on a set $X$ when size of $X$ equals two. It is clear that the discrete metric is one possible metric. But is the only requirement that the metric assign different positive real number to $d(x,y)$ and zero to $d(x,x)$? And if so, is

$d(x,y) = r$,

$d(x,x) = 0$

a metric for every possible $r \in \mathbb R_{> 0}$?

The other case I am thinking about is $|X|=1$. Are there any other metric apart from $d(x,x) = 0$?

If $d(x,y)$ is a metric, then so is $ad(x,y)$ for every $a>0$. — Julien, Feb 04 '13 at 15:08

score 1 · Answer 1 · answered Feb 04 '13 at 15:10

1

On a set of two elements, the only thing to determine is $d(0,1)$. So in fact, the set of all metrics is in bijection with $\mathbb R_+^*$.

answered Feb 04 '13 at 15:10

Damien L

6,649

All metrics on set of size 2

1 Answers1