Proof That Every Second Countable Space is a Lindelof Space
Theorem
Every second countable space is a lindelof space.
Proof
A topological space
is called a Lindelof space if every open cover is reducible to a countable cover.
The theorem is equivalent to the theorem that every base
for
is reducible to a countable base for
Letbe second countable, then
has a countable base
Let
be any base for
Then for each
where 
Hence
is an open cover of
andis reducible to a finite subcover
and
is countable.
is a base forsinceis a base. Also
and
is countable.
