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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?

Thanks!

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    Do you guys know of any good book to read up on this?2012-12-14

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