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={xCasual Long Cotton Cardigan Sleeve blend Paneled ×(x)|xRl} (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 R2lL 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,dQ and αc

Thank you

Selfridge shirt t striped slogan Miss 'honey bee' Petite FWxqd00Cw
up vote 1 down vote Sleeve Cotton blend Casual Long Cardigan Paneled accepted

A closed subspace of a Lindelöf space is Lindelöf too, so if the Sorgenfrey plane is, so is L. But L in the subspace topology is discrete, as every singleton is open as [a,b)×[a,d)L={(a,a)}.

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

Your Answer

  • Long Sleeve Cardigan Cotton blend Paneled Casual
 
Selfridge queen shirt bee Miss Petite t SZwqg

By clicking "Post Your Answer", you acknowledge that you have read our updated terms of service, privacy policy and cookie policy, and that your continued use of the website is subject to these policies.

Not the answer you're looking for? Browse other questions tagged or ask your own question.