## Proof That Every closed Subspace of a Lindelof Space is a Lindelof Space

Theorem

Every closed subspace of a Lindelof space is a Lindelof space.

Proof

Supposeis a closed subspace of a Lindelof spaceandis an open cover ofEachfor someopen in

Sinceis closed,is open inhenceis an open cover of

There is a countable subcoversois a countable cover of

Henceis a countable open subcover ofandis a Lindelof space.