## Proof That Every Second Countable Space is a Lindelof Space

Theorem

Every second countable space is a lindelof space.

Proof

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

andis countable.

is a base forsinceis a base. Alsoandis countable.