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I was hoping that somebody can explain to me the definition of quotient metric spaces

I got the following definition from wikipedia:

If $M$ is a metric space with metric $d$, and $\sim$ is an equivalence relation on $M$, then we can endow the quotient set $M/{\sim}$ with the following (pseudo)metric. Given two equivalence classes $[x]$ and $[y]$, we define $$ d([x],[y]) = \inf\{d(p_1,q_1)+d(p_2,q_2)+\dotsb+d(p_{n},q_{n})\} $$ where the infimum is taken over all finite sequences $(p_1, p_2,\dots, p_n)$ and $(q_1, q_2,\dots, q_n)$ with $[p_1]=[x], [q_n]=[y],[q_i]=[p_{i+1}], i=1,2,\dots, n-1$.

From another discussion on this website I understand that we use this definition, instead of simply the infimum over d(p,q) for all possible combinations for p and q, to guarantee the triangle inequality. But it is not entirely clear to me how to (geometrically) interpret this definition and how to actually compute distances with it.

I tried to work with the following example:

$X = \{ -1,1,-2,2,1.1,2.1\}$ with $d(x,y)=|x-y|$ and $\sim\, = \{\{1,-1\},\{2,-2\},\{1.1,2.1\}\}$

and compute the distance between -1 and 1 and also the distance between -1 and 1.1.

Could somebody please be so kind to give me a step by step walk-through on how to use the definition and compute the distances for these two examples.

Thanks!

Gijs Dubbelman

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    I think it might aid in understanding (especially given the error underlying your examples) if you didn't use the same letter to denote the two metrics -- you could denote the quotient metric by $d_\sim([x],[y])$, like I did in my answer.2012-10-31

1 Answers 1