Editing Category of metric spaces (section)

==Products and functors==
The [[Product (category theory)|product]] of a finite [[Set (mathematics)|set]] of metric spaces in '''Met''' is a metric space that has the [[cartesian product]] of the spaces as its points; the distance in the product space is given by the [[supremum]] of the distances in the base spaces. That is, it is the [[product metric]] with the [[sup norm]]. However, the product of an infinite set of metric spaces may not exist, because the distances in the base spaces may not have a supremum. That is, '''Met''' is not a [[complete category]], but it is finitely complete. There is no [[coproduct (category theory)|coproduct]] in '''Met'''.

The [[forgetful functor]] '''Met''' → '''[[Category of sets|Set]]''' assigns to each metric space the underlying [[Set (mathematics)|set]] of its points, and assigns to each metric map the underlying set-theoretic function. This functor is [[Faithful functor|faithful]], and therefore '''Met''' is a [[concrete category]].