**Definition**: A space for which every open covering contains a countable subcovering is called a *Lindelöf space*

In the book *Topology* written by Munkres it is said that the Sorgenfrey plane is not Lindelöf using the next argument:

Consider the subspace L={xStripe Perfect for Striped Top Created Macy's Bar III Surplice Sleeveless ×(−x)|x∈Rl} (where Rl is the lower limit topology). It is easy to see that L is closed in R2l.

Let us cover R2l by the open set R2l−L and by all basis elements of the form

[a,b)×[−a,d)

Each of these open sets intersects L in at most one point. Since L is uncountable, no countable subcollection covers R2l

My question is, why you can't choose another form of all basis elements in order to intersect L with this basis elements such that the intersection have more than one point?

For example, why you can't choose elements of the basis with the following form?

[α,b)×[c,d) where α,b,c,d∈Q and α≠c

Thank you