Pseudometric Space - Metric Identification

The vanishing of the pseudometric induces an equivalence relation, called the metric identification, that converts the pseudometric space into a full-fledged metric space. This is done by defining if . Let and let

Then is a metric on and is a well-defined metric space.

The metric identification preserves the induced topologies. That is, a subset is open (or closed) in if and only if is open (or closed) in .