Lets $X$ be a compact metrizable space and $f:X\to Y$ be a quotient map such that $Y$ equipped with the quotient topology is Hausdorff. Thus $Y$ is metrizable. Lets $\sim$ be an equivalence relation on $X$ such that $x\sim y$ if $f(x)=f(y)$. Then for any metric $d$ compatible with the topology of $X$ one can build a (pseudo)metric $d_\sim$ on $Y$ with: \begin{equation} d_\sim(a,b) = \inf\{d(p_1,q_1) + \cdots+ d(p_n,q_n);[p_1] = a,[q_i] = [p_{i+1}],[q_n] = b\}.(1) \end{equation} Where the $\inf$ is taken over all finite chains of points $\{p_i\}_{i=1}^n$, $\{q_i\}_{i=1}^n$ between $a$ and $b$.

As in this question which has not been fully answered (Quotient of metric spaces) can we show that $d_\sim$ is a metric compatible with the quotient topology in $Y$ ?

If not, what would be a sufficient condition on the quotient map in order to have the result ?

Here is an attempt:

In Herman 1968, *Quotient of metric spaces*, in theorem 4.8, is stated the following :

THEOREM: Let $f$ be a function from a pseudo-metrizable space $X$ to a topological space $Y$, and suppose that $Y$ has the quotient topology relative to $f$, then the following are equivalent:

1) $Y$ is pseudo-metrizable

2) There exists a pseudo-metric $\rho$ compatible with the topology in $X$ such that the quotient pseudo-metric $\rho_\sim$, defined as in (1), is in fact compatible with the quotient topology of $Y$ (The definition of the quotient pseudo-metric by Herman should be equivalent to the one introduced earlier).

If $X$ is in fact metrizable, then it is pseudo-metrizable and $Y$ is also pseudo-metrizable. Therefore, from the theorem there exists a pseudo-metric $d^*$ compatible with the topology in $X$ (it is a metric as $X$ is Hausdorff) such that the quotient pseudo-metric $d^*_\sim$ is compatible with the topology in $Y$ (it is also a metric because $Y$ is Hausdorff too).

Is it possible to show that any quotient (pseudo)metric from an arbitrary metric $d$ is topologically equivalent to $d^*_\sim$ ?

We know that $X$ is metrizable and compact, thus there is a unique uniform structure in it and all metrics compatible with the topology are uniformly equivalent. Therefore any metric $d$ compatible with the topology of $X$ is uniformly equivalent to the metric $d^*$.

From uniform equivalent metrics, maybe there is a relation between their corresponding quotient pseudo-metrics but I am stucked here, do you have any idea/theorem/reference that would help me ?

Thanks