I am interested in the behavior of the quotient semi-metric on geodesic spaces, i.e. length spaces where there is always a minimal curve between two points.
I used the following definition of the quotient semi-metric: Let $(X,d)$ be a metric space and $R$ an equivalence relation on $X$. The quotient semi-metric $d_R$ is defined as
\begin{equation} d_R([x],[y]):=\inf \{\sum_{i=1}^n d(x_i,y_i) \mid x R x_1, \ y R y_n, \ x_{i+1} R y_i \ \forall i=1, \ldots n-1, n \in \mathbb{N} \} \end{equation}
This construction in general yields a semi-metric space $(X/R,d_R)$. One can then identify points of vanishing $d_R$-distance in $X/R$ to obtain a metric space which I will denote by $X/d_R$.
In several sources (e.g. Burago, Burago, Ivanov - A course in metric geometry or Bridson, Haeflinger - Metric spaces of non-positive curvature) there is a result that states:
If $(X,d)$ is a length space, then $X/d_R$ is also a length space. In other words if I start with a length space and then identify points of zero distance in the quotient semi-metric I end up with a length space again.
My intuition is that if I start with a geodesic space then the resulting length space need not be a geodesic space. My question then is if there are certain assumptions I can impose on my geodesic space (or on the equivalence relation?) such that after this construction I obtain again a geodesic space? I am grateful not only for answers but also for references! Also a nice/intuitive counterexample would be appreciated!