Sorgenfrey line

Remarks

The rationals are dense, since each $[a, b)$ contains a rational number. The intervals $[x, x + 1/n)$ for $n \ge 1$ form a countable neighbourhood base at $x$. Let $\mathcal{B}$ be any base. For each $x \in \R$ choose $B_x \in \mathcal{B}$ with $x \in B_x \subseteq [x, x + 1)$. Then $x$ is the least element of $B_x$, so $x \mapsto B_x$ is injective and $\mathcal{B}$ is uncountable. A separable metrizable space is second-countable, so $\R_\ell$ is not metrizable. The Sorgenfrey line is finer than the usual topology, it is Lindelöf and normal, and it is totally disconnected because each $[a, b)$ is both open and closed.