# Why are quotient metric spaces defined this way?

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 $[p_1]=[x], [q_n]=[y], [q_i]=[p_{i+1}], i=1,2,\dots, n-1$. In general this will only define a pseudometric, i.e. $d'([x],[y])=0$ does not necessarily imply that $[x]=[y]$. However for nice equivalence relations (e.g., those given by gluing together polyhedra along faces), it is a metric.

1. I wonder why the quotient metric is defined that way?
2. Instead, how about using the distance between two subsets of the metric space $$d'([x],[y]) = \inf\{d(p,q)\}$$ the infimum is taken over all $(p,q)$ such that $[p]=[x], [q]=[y]$?

Thanks and regards!

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In item $2$, did you mean $p$ and $q$? – joriki Mar 16 '12 at 18:49
Whoops! Thanks! – Tim Mar 16 '12 at 18:51

This is to ensure the triangle inequality. In your proposal, it can happen that $[p]$ and $[q]$ have nearby representatives and $[q]$ and $[r]$ have nearby representatives, but the two representatives of $[q]$ involved are different, so this doesn't guarantee that $[p]$ and $[r]$ have nearby representatives, so the triangle inequality may be violated.

With the Wikipedia definition, on the other hand, it's straightforward to verify the triangle inequality, since any chain of points from $[p]$ to $[q]$ and any chain of points from $[q]$ to $[r]$ can be concatenated to form a chain of points from $[p]$ to $[r]$, so the triangle inequality follows from the individual triangle inequalities. In particular, in the above situation, we can "hop" from one representative of $[q]$ to the other without extra cost.

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Do you really need the triangle inequality of $d$ to prove this? If I'm not mistaken $d'$ should be a pseudo metric even without this assumption. – Jim Mar 29 '14 at 12:42
@Jim: Great point. It follows simply because any two sequences from $p$ to $q$ and from $q$ to $r$ can be concatenated to form a sequence from $p$ to $r$. So actually this can be used to form a pseudometric from any binary function, by having each element in an equivalence class of its own. – joriki Jul 11 at 13:43

Hopefully an example will show the problem:

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

Then $$d([1],[2]) \leq d([1],[1.1]) + d([1.1],[2]),$$ but if we use any sort of infimum definition, then $$d([1],[1.1]) \leq |1-1.1| = 0.1$$ and $$d([1.1],[2])=d([2.1],[2]) \leq |2.1-2|=0.1,$$ so $$d([1],[2]) \leq 0.2 < |1-2|= 1$$

Using a supremum would have the odd effect that $d([1],[1])=d([1],[-1]) \geq 2$, so infimum seems natural, even if it is a little broken (only a pseudo-metric).

(Joriki's answer indicates the same example.)

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Clearly, using the definition from Wikipedia, you always get a smaller or equal distance than under your proposal. If the definitions disagree and your approach is employed, the triangle inequality is violated. The definition you find in Wikipedia guarantees that the triangle inequality is valid. So I will give an example in which the triangle inequality is violated under your propsal.

Let $A=\{(0,x):x\in[0,1]\}$, $C=\{(1,x):x\in[0,1]\}$, and $B=\{(x,x):x\in(0,1)\}$. Let $X=A\cup B\cup C$ and endow it with the Euclidean metric. Partition $X$ into $A$, $B$, and $C$. One can make a path from $A$ to $C$ have lenght arbitrarily close to $0$, but $d(A,C)=1$ under your proposal.

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