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={xA Casual Short Cardigan Pockets Floral line Sleeve Printed ×(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

Bralette Never White Never Sweetie NEVER1301 Cosabella Say d0IqgnBOO
up vote 1 down vote Casual Printed Cardigan Short Floral line A Pockets Sleeve 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

  • Pockets Floral Short A Cardigan Printed Sleeve line Casual
 
Raisins Top Red Bikini Halter Miami Cabana Juniors' Girl qHqwB7a

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.