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Is completeness an intrinsic property of a space that is independent of metric? For example, since $\mathbb{R}^n$ is complete with the Euclidean metric, is it complete with any other metric?

If completeness is an intrinsic property, why is it intrinsic?

Thanks :)

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    Nope. $\mathbb R^n$ is homeomorphic to an open ball, and with the Euclidean metric on the ball, it is not complete.2012-02-29
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    You should really specify if you mean "any other metric" or "any other metric that generates the same topology."2012-02-29
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    Sorry, but actually that's something else I didn't know. I had always assumed different metrics would generate different topologies, but I see what you mean, e.g. if we take $d(x,y) = |x-y|$ and $d(x,y) = 2|x-y|$ they generate the same topology.2012-02-29

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