Theorem
Every closed subspace of a Lindelof space is a Lindelof space.
Proof
Suppose
is a closed subspace of a Lindelof space
and
is an open cover of
Each
for some
open in
Sinceis closed,
is open in
hence
is an open cover of
There is a countable subcover
so
is a countable cover of
Hence
is a countable open subcover of
andis a Lindelof space.