Proof That a Discrete Space is Separable if and Only if it is Countable
Theorem
A discrete space
is separable if and only if it is countable.
Proof
Letbe a space with the discrete topology. Every subset of
is both open and closed so the only dense subset ofisitself. Hencecontains a countable dense subset if and only ifis countable. This means that any discrete spaceis separable if and only if is countable.
Alsowith the cofinte topology is separable. Supposeis countable thenis a countable dense subset of
Suppose then thatis not countable, thencontains a non - finite countable subset
In the cofinite topology the only closed sets are the finite sets and 
Hence the closure of
is the entire spaceso
Sinceis countableis
