 Quotient metric

Let $(X,d)$ be a metric space and let $Q=X/\sim$ be a quotient set by an equivalence relation. The chain construction gives the largest pseudometric $d_Q$ on $Q$ for which the canonical projection of a quotient set $q:X\to Q$ is a non-expansive map. For classes $A,B$, putEach chain pays for movement inside $X$ and jumps at no cost between members of the same equivalence class. Reversing and concatenating chains proves symmetry and the triangle inequality. Taking a single link gives $d_Q(qx,qy)\le d(x,y)$. Conversely, any pseudometric making $q$ a non-expansive map is bounded above by the cost of every chain, hence by this infimum.

The result need not be a metric on the original quotient set. In the subset $X=\{0\}\cup \{1/n:n\ge 1\}$ of the real line, with its induced metric, collapse all the points $1/n$ to one class $A$, keeping $\{0\}$ separate. Then $d_Q(\{0\},A)\le 1/n$ for every positive integer $n$, so these distinct classes have distance zero. Identifying all zero-distance pairs produces the metric quotient of a pseudometric, on which the induced distance is a metric.