What is the completion of a metric space $(\mathbb{Q}, |\ \ |)$?
What is the completion of a metric space $(\mathbb{Q}, |\ \ |)$?
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metric-spaces
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0Do you know why $\mathbb{Q}$ is not complete? – 2012-06-10
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0it's $\mathbb{R}$. – 2012-06-10
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0I'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. – 2012-06-10
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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. – 2012-06-10
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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. – 2012-06-10
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0is the usual metric, I have problems to show the isomorphism – 2012-06-10
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0@david : What isomorphism? Between $\mathbb R$ and the completion? Are you thinking about an isomorphism of metric spaces? (i.e. an isometry) – 2012-06-10
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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. – 2012-06-10