Proof That a Countably Compact Metric Space is Separable
Any countably compact metric spaceis separable.
Letbe any positive number. There is a maximal subsetsuch that for
Supposeis infinite for somethenhas an accumulation point
contains infinitely many points ofAll points inare a distance less thanfromsocontains infinitely many points ofThis is a contradiction andis finite for each
Then for anyelse contradicting the maximality of
For each positive integer n definethenis a countable dense subset ofandis separable.