Proof That Every Second Countable Space is a Lindelof Space
Every second countable space is a lindelof space.
A topological spaceis called a Lindelof space if every open cover is reducible to a countable cover.
The theorem is equivalent to the theorem that every baseforis reducible to a countable base forLetbe second countable, thenhas a countable base
Letbe any base forThen for eachwhere Henceis an open cover ofandis reducible to a finite subcover
is a base forsinceis a base. Alsoandis countable.