I was reading a note on the Sorgenfrey line at this web page.
In the first proof (proof A), the Sorgenfrey line is Lindelöf. So my question is, with the same structure, is $[0,1)$ with the sorgenfrey topology also Lindelöf?
Yes, $[0,1)$ with the Sorgenfrey topology is Lindelöf, and the proof in Dan Ma’s Topology Blog works just as well. In fact, the Sorgenfrey line $S$ is hereditarily Lindelöf, and the same proof works; this is result B of the blog post.