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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 :)

Solaris
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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. – Cheerful 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

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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 $\mathbb{R}^n$.

Adrien Boulanger
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There is a property called "completely metrizable" and a space has the property if it will admit a complete metric that generates the same topology. So in that sense completeness might be regarded as an intrinsic property of the space. For example, the irrationals with the usual topology are completely metrizable. One builds the new metric by essentially killing off any Cauchy sequences that don't converge.

Patrick
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$C[a,b]$ is complete with metric $d(x,y)=\sup_{t \in [a,b]} |x(t)-y(t)|$ but incomplete with metric $d(x,y)=\int_{a}^{b} |x(t)-y(t)|\,dt$. Therefore, a space might be complete or incomplete depending on the metric. Whereas using Bolzano-Weierstrass Theorem, one can prove that every finite dimensional normed space will always be complete. Therefore every finite dimensional normed linear space is complete irrespective of the norm used over them.

Oogway
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