TITLE: Find Metric in $\mathbb{R}^2$ s.t. it is not Complete
QUESTION [5 upvotes]: My friend ask me: How to define a metric in $\mathbb{R}^2$ in such an way that $\mathbb{R}^2$ is not complete. I gave him the following metric:
Let $B=\{x\in\mathbb{R}^2:\ \|x\|<1\}$. By a diffeomorphism we can think that $\mathbb{R}^2$ is $B$. In this way we have that the points in $B$, close to the boundary of $B$, are the points in $\mathbb{R}^2$ with big norm in $\mathbb{R}^2$. Hence, if $F:\mathbb{R}^2\rightarrow B$ is the diffeomorphism, we can define the metric in $\mathbb{R}^2$ by $$d(x,y)=\overline{d}(F(x),F(y))$$
where $\overline{d}$ is the euclidean metric restricted to $B$.
He liked the metric, but he asked me an more "elementary" metric, not so trivial but not so elaborated.
In this way, can you guys please help me to find more metrics?
Thanks

REPLY [6 votes]: Since everybody tries something almost everywhere smooth, I repeat my non-smooth solution from the comment above as an asnwer:
The set $\mathbb R^2$ as the same cardinality as $\mathbb R\setminus \mathbb Q$. Therefore there exists a bijection $F\colon \mathbb R^2\to \mathbb R\setminus \mathbb Q$.
Then we can define the metric
$$d(x,y)=|F(x)-F(y)|$$
on $\mathbb R^2$, which makes it an incomplete metric space, of course isomorphic to $ \mathbb R\setminus \mathbb Q$ with standard metric.