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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.

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    It is a complete metric space. http://en.wikipedia.org/wiki/Complete_metric_space#Examples2011-03-21

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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.

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    See also [Real A$n$alysis in Reverse](http://arxiv.org/abs/1204.4483).2013-06-27