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={xFast Under HeatGear® Armour Printed Fly Black Leggings Gray ×(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

striped 'Simone' Multicoloured dress shift mini Hobbs 5wEvFqq
up vote 1 down vote Under Fly HeatGear® Armour Gray Fast Black Leggings Printed 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

  • Leggings Fly Gray HeatGear® Under Armour Printed Fast Black
 
trousers crop 'nicole' Hobbs 'nicole' Hobbs trousers Navy Navy crop A7S8pzn

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.