I have the incomplete metric space $M = (\mathbb{R}, d)$, where the metric $d$ is given by $d=|\tanh(x)-\tanh(y)|$.

I have the following definition of a completion:

"Given an incomplete metric space $(X, d)$, a completion is a pair $((X', d'), \Phi)$, where $(X', d')$ is a complete metric space and $\Phi$ is an isometry between $(X, d)$ and $(X', d')$, sucht that $\Phi(x)$ is dense in $X$."

I know already that ${a_n}=n$ is a Cauchy sequence in $M$, but does not converge in $M$, since $\infty \notin \mathbb{R}$ (hence $M$ is not complete).

So how exactly can I find the completion of $M$?

  • 1
    $\begingroup$ Consider $\tanh \colon (\mathbb{R},d) \to (\mathbb{R}, \lvert\,\cdot\,\rvert)$. What do you know about that map? $\endgroup$ – Daniel Fischer Nov 11 '17 at 14:39
  • 1
    $\begingroup$ For a general strategy that fits well with the answer of @HagenvonEitzen, see math.stackexchange.com/questions/2469018/… $\endgroup$ – Lee Mosher Nov 11 '17 at 14:46

Instead of adding points to $\Bbb R$, simply note that $x\mapsto \tanh x$ is by definition an isometry $(\Bbb R,d)\to ((-1,1),d_0)$ where $d_0$ is the standard metric, $d_0(x,y)=|x-y|$. This suggests that $(([-1,1],d_0),\tanh)$ is a good candidate ...

  • $\begingroup$ This helps a lot! Thank you! $\endgroup$ – Philipp Nov 11 '17 at 15:58

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.