As an example of topological space, is there a completion of $\mathbb{Q}$, other than $\mathbb{R}$ and p-adic number $\mathbb{Q}_p$, in use?

In other words, is there other kind of metric defined on $\mathbb{Q}$ with some interesting properties?

  • 9
    $\begingroup$ $\mathbb{Q}$ is a pretty horrible topological space and you can do all sorts or horrible things with it. For example, split it into disjoint parts $(-\infty, \sqrt{2}) \cap \mathbb{Q}$ and $(\sqrt{2}, \infty) \cap \mathbb{Q}$, and complete the two pieces in different ways. $\endgroup$
    – user14972
    Commented Jan 5, 2018 at 2:30
  • 5
    $\begingroup$ en.wikipedia.org/wiki/Ostrowski%27s_theorem $\endgroup$
    – Mr. Chip
    Commented Jan 5, 2018 at 2:31
  • 6
    $\begingroup$ There are three different absolute values that can be put on $\mathbb{Q}$ then completed to give intrinsically different spaces: the usual absolute value (which completes to $\mathbb{R}$, a $p$-adic absolute value (which completes to $\mathbb{Q}_p$, or the trivial absolute value (which makes $\mathbb{Q}$ into a complete, totally disconnected metric space). This is essentially the substance of Ostrowski's theorem. That said, if you don't require your metric to come from an absolute value, you can do almost anything. $\endgroup$
    – Xander Henderson
    Commented Jan 5, 2018 at 2:32
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    $\begingroup$ The completion is not as a topological space. There is no notion of a "complete topological space". You need more structure to talk about completeness, so you can talk about uniform spaces, etc. In this particular case you mean metric space. $\endgroup$
    – Asaf Karagila
    Commented Jan 5, 2018 at 3:50
  • $\begingroup$ @Hurkyl I've never seen that before. That's really really gross. $\endgroup$ Commented Jan 5, 2018 at 3:50


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