Exercise 3.25 Let $X$ be a metric space whose points are the rational numbers, with metric $d(x,y) = |x-y|$. What is the completion of this space?
Since $\mathbb{Q}$ is dense in $\mathbb{R}$ - which is complete - my take is that the completion of the space described in the exercise is in fact $\mathbb{R}$. The problem is this "answer" doesn't even mention the metric proposed by the exercise. I don't think it's relevant because, well, this is the metric of $\mathbb{R}$, but I think I'm missing something here.
That's right: $\Bbb R$ is the completion of $\Bbb Q$ under the standard metric. The only thing you might be missing here is that the metric $d$ on $\Bbb Q$ is the restriction of the usual metric on $\Bbb R$, under which $\Bbb R$ is known to be complete. In general if you have metric spaces $X\subset Y$ with $Y$ complete, and $X$ dense in $Y$, then $Y$ is the completion of $X$ (as long as the metric of $X$ is the restriction of the metric of $Y$).