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Suppose that we have an equivalence relation $\sim$ in a topological metrizable space $(X,d).$ Then we can endow $X/\sim$ with the quotient topolgy.

Also, under certains circunstances, there exists a quotient metric in $X/\sim$ given by

$$d'([x],[y]) = \inf \left \{ d(p_1,q_1) + d(p_2,q_2) + ... + d(p_n,q_n)\right \},$$

where $[p_1] = [x], [q_n] = [y],$ and $[q_i] = [p_{i+1}].$

I wonder under which circuntances, the quotient topology on $X/\sim,$ coincides with the topology induced by $d'.$

We can assume (if necessary) that the function $X\ni x\to [x]\in X/\sim$ is open.

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