Is Completeness intrinsic to a space?

analysis convergence-divergence metric-spaces

Solution 1:

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 $\mathbb{R}^n$.

Solution 2:

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.

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