general topology – Topologies on $mathbb{R}$

I have found this in the internet:

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I already know that
$mathcal{B}_1$ is the Sorgenfrey line. What I did not know is that $b$ doesn’t need to be real in order to get the line topology. How can i show this?
I have found that $mathcal{B}_2$ is also a Topology (on $mathbb{R}$?) but is not equivalent with the Sorgenfrey line. I cannot find anywhere proofs for this and I don’t have an idea how to show this.

Has anyone a proof for this two statements?

In general: Is there somewhere an overview for all the topologies on $mathbb{R}$ generated by all the intervals?