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}.
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