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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 $[p_1]=[x], [q_n]=[y],[q_i]=[p_{i+1}], i=1,2,\dots, n-1$.

I wonder when does this define a pseudo metric and when does it define a metric? I cannot see how this can define a pseudo metric because the equivalence relationship partitions $M$ in equivalence classes which are disjoint sets in $M$ and $d(,)$ is a metric for $M$. What I am missing? Is there a good text book to read up on this?


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Have you actually looked at any examples? Or tried to turn your idea into an actual proof? – Chris Eagle Dec 13 '12 at 16:45
Look at the example of $M = \mathbb{R^2}$ with the Euclidean metric, and the equivalence relation being $(x_1,y_1) \sim (x_2,y_2)$ when $x_1y_1= x_2y_2$ (one equivalence class is the unions of the coordinate axes, the other equivalence classes are hyperbolas). – Omar Antolín-Camarena Dec 13 '12 at 17:46
Partition $\mathbb R$ as $(-\infty,0]$ and $(0,\infty)$. What's the distance between the two equivalence classes? – JSchlather Dec 13 '12 at 17:47
Thanks Chris, Omar and Jacob! – Gijs Dubbelmanm Dec 14 '12 at 9:04
Do you guys know of any good book to read up on this? – Gijs Dubbelmanm Dec 14 '12 at 9:06

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