The lower limit topology on the real line $\mathbb{R}_{\ell}$ is not metrizable. However, $\displaystyle \tilde{\mathbb{R}_{\ell}}$ the rational lower limit topology on $\displaystyle \mathbb{R}$ generated by basis: $\displaystyle [q_1,q_2)$ where, $\displaystyle q_1,q_2 \in \mathbb{Q}$ is metrizable (it's regular + 2nd countable, hence the result follows by Urysohn's metrization theorem). It's interesting to actually write down the metric one gets from Urysohn's theorem.

The key idea is to observe that for any enumeration $\{q_n\}_{n \in \mathbb{N}}$ of the rationals $\mathbb{Q}$, the countable family of continuous function $\displaystyle f_n(x) := \begin{cases} 1 & \text{ if } x \ge q_n \\ 0 & \text{ otherwise }\end{cases}$ (continuous in $\displaystyle \tilde{\mathbb{R}_{\ell}}$ topology) separates disjoint basic clopen neighborhoods.

Hence, the evaluation map $\displaystyle f: \tilde{\mathbb{R}_{\ell}} \to \{0,1\}^{\omega}$ given by $\displaystyle f(x) = (f_n(x))_{n \in \mathbb{N}}$ is an imbedding into the Cantor space $\displaystyle \{0,1\}^{\omega}$ with product topology (as it separates points and closed sets).

Thus the pullback of the metric on Cantor space v.i.a. $f$-imbedding gives us an explicit metric for $\displaystyle \tilde{\mathbb{R}_{\ell}}$: $$d(x,y) = \sum\limits_{n=1}^{\infty} \frac{|f_n(x) - f_n(y)|}{2^n} = \begin{cases} \sum\limits_{x < q_n \le y} \frac{1}{2^n} & \text{ when } x < y \\ 0 & \text{ when } x = y.\end{cases}$$

The function $\displaystyle g(x) = \sum\limits_{q_n \le x} \frac{1}{2^n}$ is monotone increasing and precisely continuous at irrationals and only right continuous at rational points in the standard $\mathbb{R}$ topology as one would expect, and has jumps of magnitude $\displaystyle \frac{1}{2^n}$ at the point $\displaystyle x = q_n$ for each $n \in \mathbb{N}$. The metric can be written as $\displaystyle d(x,y) = |g(x) - g(y)|$.

The underlying "big gun" that would connect both the metrics on $\mathbb{Q}_\ell$ exhibited above would be the following theorem due to Sierpinski:

*(Source: Our Prof asked this problem during a Topology lecture in 2016.)*The key idea is to observe that for any enumeration $\{q_n\}_{n \in \mathbb{N}}$ of the rationals $\mathbb{Q}$, the countable family of continuous function $\displaystyle f_n(x) := \begin{cases} 1 & \text{ if } x \ge q_n \\ 0 & \text{ otherwise }\end{cases}$ (continuous in $\displaystyle \tilde{\mathbb{R}_{\ell}}$ topology) separates disjoint basic clopen neighborhoods.

Hence, the evaluation map $\displaystyle f: \tilde{\mathbb{R}_{\ell}} \to \{0,1\}^{\omega}$ given by $\displaystyle f(x) = (f_n(x))_{n \in \mathbb{N}}$ is an imbedding into the Cantor space $\displaystyle \{0,1\}^{\omega}$ with product topology (as it separates points and closed sets).

Thus the pullback of the metric on Cantor space v.i.a. $f$-imbedding gives us an explicit metric for $\displaystyle \tilde{\mathbb{R}_{\ell}}$: $$d(x,y) = \sum\limits_{n=1}^{\infty} \frac{|f_n(x) - f_n(y)|}{2^n} = \begin{cases} \sum\limits_{x < q_n \le y} \frac{1}{2^n} & \text{ when } x < y \\ 0 & \text{ when } x = y.\end{cases}$$

The function $\displaystyle g(x) = \sum\limits_{q_n \le x} \frac{1}{2^n}$ is monotone increasing and precisely continuous at irrationals and only right continuous at rational points in the standard $\mathbb{R}$ topology as one would expect, and has jumps of magnitude $\displaystyle \frac{1}{2^n}$ at the point $\displaystyle x = q_n$ for each $n \in \mathbb{N}$. The metric can be written as $\displaystyle d(x,y) = |g(x) - g(y)|$.

*Update (06.11.2019):***I recently came across this interesting post in stack****What is a metric for $Q$ in the lower limit topology?**where the following interesting metric was proposed for $\mathbb{Q}_\ell$ (the rational lower limit topology):Write your rationals as "mixed fractions," that is as their integer floor plus a fractional part and defineCompared to the jump function $g$ mentioned earlier this metric seems to be modeled on the discontinuities of the

$$d\left(a\frac{p}{q},b\frac{r}{s}\right)=\begin{cases}|a-b|,& a\neq b\\ d'\left(\frac{p}{q},\frac{r}{s}\right), &\textrm{otherwise} \end{cases}.$$

Define the distance between pure fractions separately. Assume WLOG that $\frac{p}{q}\leq \frac{r}{s}$. Then set $$d'\left(\frac{p}{q},\frac{r}{s}\right)=\max\left(\left|\frac{p}{q}-\frac{r}{s}\right|,\frac{1}{m}\right), \frac{k}{m} \in \left(\frac{p}{q},\frac{r}{s}\right].$$

**Thomae's function**. To elaborate this point let us consider the separating family of functions $(f_r)_{r \in \mathbb{Q}}$ to be $$f_r(x) := \begin{cases} 1/q & \text{ if } x \ge r \\ 0 & \text{otherwise} \end{cases}$$ where, $r = p/q \in \mathbb{Q}$ with $(p,q) = 1$. For a fixed $r \in \mathbb{Q}$ the metric $$d_r(x,y) := \begin{cases} |x-y| & \text{ if } x,y \ge r \text{ or } x,y \le r \\ \max\left(|x-y|, 1/q\right) & \text{ if } x < r \le y\end{cases}$$ precisely makes $f_r$ continuous. Modifying the above metric to incorporate jumps at all $r \in \mathbb{Q}$ which has denominator $q$ in irreducible representation followed by a factor $q$ scaling we may consider $$d_q^\prime(x,y) := \begin{cases} q|x-y| & \text{ if } x,y \in \left[\frac{p-1}{q},\frac{p}{q}\right) \text{ for some } p \in \mathbb{Z} \\ \max\left(q|x-y|, 1\right) & \text{ if } x < \frac{p}{q} \le y \end{cases}$$ which now precisely makes those $f_r$'s continuous s.t., denominator of $r \in \mathbb{Q}$ in irreducible representation is $q$. Then the product metric constructed from these $\{d_q'\}_{q \in \mathbb{N}}$ viz. $$d' := \sup_{q \in \mathbb{N}} \left(\frac{\max(d_q',1)}{q}\right)$$ gives a metric compatible to $\mathbb{Q}_{\ell}$ equivalent to the one proposed in the post. Then Thomae's function models the 'jump set' of the metric $d'$.The underlying "big gun" that would connect both the metrics on $\mathbb{Q}_\ell$ exhibited above would be the following theorem due to Sierpinski:

Both the metrics $d$ and $d'$ above are constructed from explicit homeomorphisms of $\mathbb{Q}_\ell$ with $\mathbb{Q}$.Theorem(Sierpinski, 1920). A dense in itself countable metric space is homeomorphic to $Q$.