Metric space

Set

context $X$ … set
definiendum $\langle X,d\rangle\in\mathrm{it}$
postulate $d$ … metric $X$

We can reconstruct the set underlying a metric via $\text{dom}(\text{dom}(d))=\text{dom}(X\times X)=X$, so the set of metrics and the set of metric spaces over $X$ are in bijection.

Reference

Wikipedia: Metric space


Requirements*

Metric

Refinement of

Hausdorff space

Equivalent to

Metric