A useful source of counterexamples in topology is $\mathbb R_\ell$, the set $\mathbb R$ together with the lower limit topology generated by half-open intervals of the form $[a,b)$. For example this space is separable, Lindelöf and first countable, but not second countable. So in particular, this implies that $\mathbb R_\ell$ is not metrizable.
Now Urysohn's metrization theorem says that any regular, second countable space is metrizable. Applying this to $\mathbb Q_\ell\subset \mathbb R_\ell$ gives the (for me rather counterintuitive) fact that there must exist a metric on $\mathbb Q$ inducing the lower limit topology. Which leads to the question:
What is a concrete example of a metric inducing the lower limit topology on $\mathbb Q_\ell$?
It certainly should be possible to go through the steps of the Urysohn metrization theorem to give a (semi-)concrete embedding of $\mathbb Q_\ell$ in $\mathbb R^\omega$ (for instance picking an enumeration of the rationals and constructing functions which separate points from closed sets, then look at the cartesian product of these maps) and then give the metric on $\mathbb Q_\ell$ in terms of this embedding.
But this is really not what I'm looking for, here. I'd be much more interested in a simple metric for $\mathbb Q_\ell$.
Because "simple" is not well-defined, I will definitely also welcome answers which exhibit a metric, but which I would not consider to be simple. On the other hand, if you see a reason for why there may just be no "simple" metrics, please feel free to point this out as well.
Thanks!
Best Answer
Let $\nu:\mathbb{Q} \to \mathbb{N}$ be an enumeration of $\mathbb{Q}$. Then $\displaystyle d(x,y):=\sum_{\min(x,y) < r \le \max(x,y)} 2^{-\nu(r)}$ will do.