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Why is it that $\mathbb{Q}$ cannot be homeomorphic to any complete metric space?

Certainly $\mathbb{Q}$ is not a complete metric space. But completeness is not a topological invariant, so why is the above statement true?

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    Don't forget to accept the answer you find most useful (if there is one); click on the checkmark on the left of the answer. If you think none of the answers are enough, then you might want to ask further questions so they can be clarified.2010-12-29

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