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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 (where is the lower limit topology). It is easy to see that is closed in .

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

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

My question is, why you can't choose another form of all basis elements in order to intersect 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?

Thank you

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A closed subspace of a Lindelöf space is Lindelöf too, so if the Sorgenfrey plane is, so is . But in the subspace topology is discrete, as every singleton is open as .

A discrete space is Lindelöf iff is is countable (take the open cover by singletons). is not countable, contradiction.