## Proof That a Discrete Space is Lindelof if and Only if it is Countable

Theorem

A discrete spaceis Lindelof if and only if it is countable.

Proof

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.