Equivalence of two metrics defined on $\mathbb{R}^2$

Question

The two metrics $d_{1}$ and $d_{2}$ are said to be topologically equivalent if they generate the same topology.

Suppose $d_1(x,y)=\sqrt{(x_1-y_1 )^2+(x_2-y_2 )^2}$ (euclidean distance)

$d_2(x,y) = \left\{ \begin{array}{ll} |y_1-y_2 | & \mbox{if } x_1 = x_2 \\ |y_1 |+|y_2 |+|x_1-x_2 | & \mbox{if } x_1 \neq x_2 \end{array} \right.$

How I can conclude whether $d_1$ is equivalent to $d_2$ or not ?

Normally you look at the $\epsilon$-balls. – Leo Jan 13 '14 at 22:14 — Leo, Jan 13 '14 at 22:14

score 2 · Accepted Answer · edited Jan 13 '14 at 22:16

2

The sequence $(\frac1n,1)$ converges to $(0,1)$ with respect to $d_1$, but not with respect to $d_2$ because $$ d_2\left((\tfrac1n,1),(0,1)\right)=2+\frac1n.$$

edited Jan 13 '14 at 22:16

Did

279,727

answered Jan 13 '14 at 22:15

Hagen von Eitzen

374,180

thanks for your answer. Here you have proved that they are not equivalent but how I can prove that two metrics are equivalent? It would be very useful for me if you have example. – Motasem M. Al-wazir Jan 13 '14 at 22:27

score 0 · Answer 2 · answered Jan 13 '14 at 22:23

0

Take $$x = \left(1,1\right)^T \text{ and } y=\left(2,2\right)^T$$

Then $$d_2(x,y) = 0$$ but $$x\neq y$$

Thus $d_2$ is not even a metric on $\mathbb{R}^2$.

Are you sure you wrote down the task correctly?

answered Jan 13 '14 at 22:23

user127.0.0.1

7,267

the two points are like this (x1,y1) and (x2.y2) so for x=(1,1) and y=(2,2). d2(x,y)= 4 not 0 as you said . please review your answer. – Motasem M. Al-wazir Jan 13 '14 at 22:27
2

@MotasemM.Al-wazir Then $d_1$ is not a metric with the same argument – user127.0.0.1 Jan 13 '14 at 22:34
both of them are metrics . please check the three conditions for metric – Motasem M. Al-wazir Jan 14 '14 at 10:25
2

No, they are not. Check the definitions in your question, there is something wrong (maybe just a typo), – user127.0.0.1 Jan 14 '14 at 12:12
thanks for you interest but the Hagen's answer is correct for the question – Motasem M. Al-wazir Jan 14 '14 at 17:43

Equivalence of two metrics defined on $\mathbb{R}^2$

2 Answers2