What is the completion of a metric space $(\mathbb{Q}, |\ \ |)$?
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Do you know why $\mathbb{Q}$ is not complete? – tomcuchta Jun 10 '12 at 16:37
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it's $\mathbb{R}$. – Albert Jun 10 '12 at 16:38
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I'm pretty sure if you just looked this up on Wikipedia you'd find it. If you're having trouble with that definition, feel free to ask about the details. – Ben Millwood Jun 10 '12 at 16:40
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1@Glougloubarbaki : It depends on what "$| , |$" means. It can be $\mathbb R$ or $\mathbb Q_p$, the $p$-adics. It all depends on the chosen metric. – Patrick Da Silva Jun 10 '12 at 16:42
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2@david : Perhaps you should precise what "$| , , |$" means. If it means the standard absolute value (the geometric distance between two points), then your completion you're looking for is $\mathbb R$, because it can precisely be defined like this. If you want details, as benmachine said, just ask. – Patrick Da Silva Jun 10 '12 at 16:44
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is the usual metric, I have problems to show the isomorphism – david Jun 10 '12 at 16:44
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@david : What isomorphism? Between $\mathbb R$ and the completion? Are you thinking about an isomorphism of metric spaces? (i.e. an isometry) – Patrick Da Silva Jun 10 '12 at 16:45
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1"a problem to show the isomorphism" ... OK, on one side is the completion of $\mathbb Q$, on the other side is $\mathbb R$ ... so we need a definition of $\mathbb R$ in order to help you. – GEdgar Jun 10 '12 at 16:50
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A metric space $X$ is complete if every Cauchy sequence $a_n \in X$ converges to an element $a \in X$.
The completion of $X$ therefore is the metric space $\bar X$, that contains all elements of $X$, plus the limits of all possible cauchy sequences in $X$ equipped with the same metric as $X$. There is no straight forward way in finding the completion of a Metric space.
In your particular case it was already mentioned in the comments, that the completion of $\mathbb{Q}$ with the canonical metric is $\mathbb{R}$ with the canonical metric.
Haatschii
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If $X$ is a metric space, then the completion of $X$, denote it by $X_c$, is the smallest complete metric space containing $X$ as a subspace. That is, if $Y$ is complete and contains $X$ as a subspace, then $Y$ also contains $X_c$ as a subspace.
If $Y$ is complete, then a subspace of $Y$ is complete if and only if it is closed in $Y$.
Assuming that you have proved these two things, you can use them to find that the completion of $(\mathbb{Q}, |\cdot|)$ is $(\mathbb{R}, |\cdot|)$. What must a complete subspace of $\mathbb{R}$ containing $\mathbb{Q}$ be?
Mikko Korhonen
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