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.