TITLE: What kind of completeness is the completeness of $\mathbb{R}$?
QUESTION [8 upvotes]: As opposed to the algebraic completion of $\mathbb{Q}$, which yields the algebraic numbers, we can say that $\mathbb{R}$ is complete in the sense that every non-empty subset of $\mathbb{R}$ bounded by above has a supremum.
So, it isn't algebraically complete, but is it topologically or metrically complete? What would be the right word to describe its completeness?
Thanks.

REPLY [9 votes]: The reals are complete as a metric space (http://en.wikipedia.org/wiki/Real_number#Completeness) and as an ordered set in the sense of Dedekind (http://en.wikipedia.org/wiki/Dedekind_completion), and also categorically as the unique complete Archimedean ordered field.