Proof That a Countably Compact Metric Space is Separable
Theorem
Any countably compact metric space
is separable.
Proof
Let
be any positive number. There is a maximal subset
such that for
Suppose
is infinite for some
thenhas an accumulation point
contains infinitely many points of
All points inare a distance less thanfrom
socontains infinitely many points of
This is a contradiction andis finite for each
Then for any
else contradicting the maximality of
For each positive integer n define
then
is a countable dense subset of
andis separable.
