Proof That a Discrete Space is Separable if and Only if it is Countable
A discrete spaceis separable if and only if it is countable.
Letbe a space with the discrete topology. Every subset ofis 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 ofSuppose then thatis not countable, thencontains a non - finite countable subsetIn the cofinite topology the only closed sets are the finite sets and Hence the closure ofis the entire spaceso