1
$\begingroup$

What is the completion of a metric space $(\mathbb{Q}, |\ \ |)$?

  • 0
    Do you know why $\mathbb{Q}$ is not complete?2012-06-10
  • 0
    it's $\mathbb{R}$.2012-06-10
  • 0
    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.2012-06-10
  • 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
  • 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
  • 0
    is the usual metric, I have problems to show the isomorphism2012-06-10
  • 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
  • 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

2 Answers 2