Metric group

Definition
A metric group is a topological group whose underlying topological space is a metrizable space, i.e., there exists a metric giving that topology.

No compatibility requirement
We do not require that the left and right multiplication maps be isometries with respect to the metric. However, in the case of a metric group that is also a compact group, we can use averaging over the Haar measure to replace the metric by a new metric such that the left and right multiplication maps by all elements are isometries with respect to that metric. In the non-compact case, we may or may not be able to replace the metric in this fashion.