# quotient metric spaces for dummies

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. – joriki Oct 31 '12 at 21:53

Your examples are fundamentally flawed in that you're asking for the distance between points of $X$, but these aren't points of $X/\sim$, so you can't compute their distance in the quotient metric $d_\sim$. You can ask what the distance from $[-1]$ to $[1]$ is, and the answer is $0$, since these are the same points (equivalence classes) of $X/\sim$. For the second one, you can ask for the distance from $[-1]$ to $[1.1]$. To find this, enter the teleporter at $-1$, jump to $1$ for free, and walk to $1.1$ by foot, for a total distance $d_\sim([-1],[1.1])=d_\sim([1],[1.1])=d(1,1.1)=0.1$.
@Gijs: You're welcome! No, for $d([1],[2])$, the optimal teleportation sequence is to walk to $1.1$, then teleport to $2.1$, then walk to $2$ for a total walking distance of $0.2$. In fact this is a good example for understanding how the definition guarantees the triangle inequality. – joriki Oct 31 '12 at 21:57