Let $(X,\tau,d)$ be a space where $\tau$ is a topology and $d$ is a metric, where the topology $\tau$ is not necessarily compatible with $d$.


Is there a canonical name for such a structure (maybe under some extra assumption on the relation between the metric and the topology)?


For example, assuming that all closed balls $\overline{B}(x,\epsilon)=\{y\in X\mid d(x,y)\leq \epsilon\}$ define with the metric are also closed in the topology $(X,\tau)$.


**Edit:** I will give some more context accordingly to the comments, since the question as stated right now might be a bit too general.


Spaces with two topologies $(X,\tau,\sigma)$ are called bitopological spaces, and there is a lot of theory behind them.
I recall from a conversation that if the second topology $\sigma$ comes from a metric $d$ and the metric interacts "nicely" with the topology, this structure $(X,\tau,d)$ goes under a different name and that this notion has a wide literature as well.


I was hoping to get some information about this type of structure.



Name of a space with both a topology and a metric that are not compatible?
A "topometric space" is a triple $(X,\tau,\rho)$ where $\tau$ is a topology on $X$ and $\rho$ is a metric on $X$ which is lower semi-continuous with respect to the topology. These are useful in continuous logic. More details can be found in papers by Ben Yaacov.


