TITLE: Cardinality of cofinite topology
QUESTION [5 upvotes]: Does it make sense to ask about cardinality of closed finite topology defined on the set of Natural Numbers? Is it countable or uncountable? Is it possible to prove that it is countable? 
Further, if it countable, is every closed finite topology (cofinite topology) defined on countable set again countable?

REPLY [5 votes]: First note that the map $X\mapsto\Bbb N\setminus X$ is a bijection between the non-trivial cofinite open sets, and the finite open sets.
Next show that $\{A\subseteq\Bbb N\mid A\text{ is finite}\}$ is a countably infinite set.
Lastly note that if $|X|=|Y|$ then defining the cofinite topology on $X$ would be easily transferable to $Y$. Hint: Every a bijection is a homeomorphism.