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.
(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):
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 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 $d'$.
$$d\left(a\frac{p}{q},b\frac{r}{s}\right)=\begin{cases}|a-b|,& a\neq b\\ d'\left(\frac{p}{q},\frac{r}{s}\right), &\textrm{otherwise} \end{cases}.$$
Define the distance between pure fractions separately. Assume WLOG that $\frac{p}{q}\leq \frac{r}{s}$. Then set $$d'\left(\frac{p}{q},\frac{r}{s}\right)=\max\left(\left|\frac{p}{q}-\frac{r}{s}\right|,\frac{1}{m}\right), \frac{k}{m} \in \left(\frac{p}{q},\frac{r}{s}\right].$$
The underlying "big gun" that would connect both the metrics on $\mathbb{Q}_\ell$ exhibited above would be the following theorem due to Sierpinski:
Theorem (Sierpinski, 1920). A dense in itself countable metric space is homeomorphic to $Q$.Both the metrics $d$ and $d'$ above are constructed from explicit homeomorphisms of $\mathbb{Q}_\ell$ with $\mathbb{Q}$.
