Proof That a Discrete Space is Lindelof if and Only if it is Countable
Theorem
A discrete space
is Lindelof if and only if it is countable.
Proof
Letbe a discrete topological space so that
is the set of all subsets of
Suppose is Lindelof then every open cover of
has a countable subcover. Each singleton set
is open.
The family of sets
is an open cover of
Sinceis Lindelof, a countable subcover
exists henceis a countable set.
Conversely supposeis countable andis discrete thenis Lindelof. Let
be an open cover ofso that
For each
exists with
and sinceis countable we can choose
Hence
s an open subcover of
andis Lindelof.
