# Questions tagged [equivalent-metrics]

In the study of metric spaces in mathematics, there are various notions of two metrics on the same underlying space being "the same", or equivalent.

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### Prove arctan metric is equivalent to euclidean metric

Given two metrics in $\mathbb{R}$, $d_1(x,y)=|\arctan x-\arctan y|$, and $d_2(x,y)=|x-y|$. I need to prove those two metrics are equivalent. I have the definition that two metrics are said to be ...
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### Proving that $\rho(x,y) = \frac{d(x,y)}{1+d(x,y)}$ is a metric and that $\rho(x,y)$ and $d(x,y)$ are equivalent metrics [duplicate]

As you can see the proof is divided into 2, the first part consists on proving that $\rho(x,y)$ is a metric My attempt i) $\rho(x,y) \geq 0$, which is clear since $d(x,y)$ is a metric, and it is $0$ ...
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### Are metrics uniformly equivalent if and only if they have the same zero-distance sets?

Let $d_1$ and $d_2$ be two metrics on the same set $X$. Then $d_1$ and $d_2$ are uniformly equivalent if the identity maps $i:(M,d_1)\rightarrow(M,d_2)$ and $i^{-1}:(M,d_2)\rightarrow(M,d_1)$ are ...
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### I have written up a proof on why norms on $\mathbb{R}^n$ are equivalent

Just wanted to share my proof with you smart people to have some feedback and to share each other's ideas. Some disclaimers: this is my first post, English is not my first language, and I know that ...
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### Existence of a metric on $\Bbb Q$ which is complete and equivalent to the usual metric.
Does there exist a metric $d$ on $\Bbb Q$ which is equivalent to the usual metric on $\Bbb Q$ such that $(\Bbb Q,d)$ is complete? I have a confusion regarding that. Because I know that equivalence of ...