TITLE: Free Metric Space?
QUESTION [5 upvotes]: Do free metric spaces exist?
Ie.: An object in the category of metric spaces and lipschitzian maps.  
If so would these be the complete metric spaces, since they satisfy a similar universal property?

REPLY [4 votes]: The free metric space on one element exists, of course. And since you are considering Lipschitz maps, it is easy to verify that the space with $n$ points and the discrete metric is freely generated by its points, when $n$ is finite. If $n$ is infinite then we have a problem: if such a metric space $X$ existed, then we would be able to construct a Lipschitz map $X \to \mathbb{R}$ where the image of the generating set is unbounded – a contradiction, since the distance between any two generating points in $X$ is finite.