= Solution
The <lower limit topology> on the <real line> is the <lower limit topology>. Every nonempty basic interval $[a,b)$ contains a rational number, so the rational numbers are a <countable> <dense subset>. By the preceding product argument, $\mathbb Q^2$ is <dense> in the square, proving its <topological separability>.
Consider the uncountable anti-diagonal
$$
D=\{(t,-t):t\in\mathbb R\}.
$$
It is a <discrete subset>: the basic rectangle $[t,t+\epsilon)\times[-t,-t+\epsilon)$ intersects $D$ only at $(t,-t)$, because both coordinates have increased weakly and their sum can remain zero only if neither increases.
It is also <closed>. At a point $(u,v)$ off the anti-diagonal, $u+v\ne0$. If this sum is positive, every sufficiently small lower-limit rectangle based at $(u,v)$ still has positive coordinate sum. If it is negative, choose $\epsilon>0$ with $2\epsilon<-(u+v)$; then $[u,u+\epsilon)\times[v,v+\epsilon)$ has negative coordinate sum throughout. These <neighborhoods> avoid $D$, so its <complement> is <open>.
If the square were <Lindelöf>, its <closed discrete subset> $D$ would be <countable> by part (c), contradicting its <bijection> with $\mathbb R$. Hence
$$
\boxed{\text{the Sorgenfrey plane is separable but not Lindelöf}.}
$$
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