Proof That a Discrete Space is Lindelof if and Only if it is Countable
A discrete spaceis Lindelof if and only if it is countable.
Letbe a discrete topological space so thatis the set of all subsets ofSuppose is Lindelof then every open cover ofhas a countable subcover. Each singleton setis open.
The family of setsis an open cover of
Sinceis Lindelof, a countable subcoverexists henceis a countable set.
Conversely supposeis countable andis discrete thenis Lindelof. Letbe an open cover ofso that
For eachexists withand sinceis countable we can choose
Hences an open subcover ofandis Lindelof.