The Sorgenfrey topology on $\mathbb{R}$ is the topology whose basic open sets are of the form $[a,b)$ where $a < b \in \mathbb{R}$.
Does it have a countable base? (I suspect not.) Certainly it is separable and first countable.
The reals with the usual topology is second countable because open intervals with rational endpoints form a countable base. However this approach does not work for the Sorgenfrey topology, since open sets of the form $[i,b)$ where $i$ is irrational, cannot be written as the union of basic open sets $[p,q)$ with rational endpoints.
