Quotient of two metrics on a non empty set $X$ is again a metric or not.

Question

My Attempt:

Case (i) If I take $d$ as $d(x,y) = \frac{d_1(x,y)}{d_2(x,y)}$ then $d$ is not well defined as $d(x,y) \geq 0$

Case(ii) If I take $d$ as $d(x,y) = \frac{d_1}{d_2}(x,y)= \begin{cases}\frac{d_1(x,y)}{d_2(x,y)}, &\text{if }x \neq y \\ 0, &\text{if }x=y. \end{cases}$ then it is well defined.

Here we see clearly that

(1) $d(x,y) \geq 0$, $\forall x,y \in X$

(2) $d(x,y) = 0 \iff x = y, \forall x,y \in X$

(3) $d(x,y) = d(y,x), \forall x,y \in X$

But I have faced a problem in triangle inequality. Please help me by giving some hints or counter examples. Thanks in advance.

score 2 · Answer 1 · answered Jul 06 '21 at 04:59

2

Hint: Consider $\Bbb R$ with discrete metric divided by standard metric.

answered Jul 06 '21 at 04:59

Hagen von Eitzen

374,180

Let $d_1$ and $d_2$ be two metrics on a non empty set $X$ then we work on $X$ at which $d$ defined. I'm saying that $d_1$ and $d_2$ are defined on same set $X$ but you let $d_1$ on $\Bbb R$ and $d_2$ on $\Bbb R^2$. – Sangam Academy Jul 06 '21 at 06:14
1

@SangamAcademy Why do you say above answer uses two sets? I see only one set $\mathbb R$ with say $d_1$ the discrete metric and $d_2$ the standard metric. – coffeemath Jul 06 '21 at 07:10
Is usual metric called standard metric ? – Sangam Academy Jul 06 '21 at 07:40
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Yes, usual also called standard. – coffeemath Jul 06 '21 at 12:58

tkf · Answer 2 · 2021-07-06T06:27:23.420

1

Consider a space consisting of 3 points: $X=\{a,b,c\}$. Let:\begin{eqnarray*}d_1(a,b)&=&d_1(b,c)=d_1(a,c)=1,\\ d_2(a,b)&=&d_2(b,c)=1,\qquad d_2(a,c)=\frac13. \end{eqnarray*}

Both $d_1,d_2$ make $X$ a metric space. However the quotient you defined does not satisfy the triangle inequality as : $1+1<3$

edited Jul 06 '21 at 06:27

answered Jul 06 '21 at 05:08

tkf

11,563

I could not understand what you say. Please elaborate it. – Sangam Academy Jul 06 '21 at 06:16
Elaborated now. – tkf Jul 06 '21 at 06:28

Quotient of two metrics on a non empty set $X$ is again a metric or not.

2 Answers2