# Is Completeness intrinsic to a space?

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. –  Grumpy Parsnip Feb 29 '12 at 12:52
You should really specify if you mean "any other metric" or "any other metric that generates the same topology." –  Arturo Magidin Feb 29 '12 at 17:59
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. –  Solaris Feb 29 '12 at 23:48

The completness depend of the metric, for instance $\mathbb{Q}$ with the standard metric is not complete, but if you consider $\mathbb{Q}$ with the distance : $d(x,y) = 0$ if $x=y$ and $d(x,y) =1$ else, $\mathbb{Q}$ is complete. You can find the same type of example in $\mathbb{R}^n$.