I stumbled over this question which is slightly different from the usual version I remember, e.g. here: in one case one extends the uniformly continuous from a metric space (not necessarily complete) to its completion, and in the other case, from a dense set to the whole space.
Are the two versions equivalent? If I remember well
- a metric space is dense in its completion
- it seems that an arbitrary metric space is not the completion of a dense subset, so it seems to me that the version with the completion is less general, but I've been confusing myself...
Remark: the usual remark... completion is a notion from metric space, density (and closure) only require a topology.
Edit: indeed a metric space that is not complete is a dense subset of itself, and it is by assumption not the completion of itself, nevertheless can there be a dense subset such that it is its completion... probably not.