# Cofinite topology and the closure

The set $$\tau=\{A \subset X, X-A$$ is finite$$\} \cup \{\emptyset\}$$ is called Cofinite Topology of $$X$$. If $$A \in \tau$$, How can I find the Closure?

I know that that the closure $$F$$ is the least closed set such that $$A \subset X$$ but if $$F$$ is closed then $$F$$ is finite or $$F=X$$ then $$F=X??$$.

Why or why not?

You're right. The only closed sets are all finite sets and $$X$$.

Assuming (to avoid trivial cases) that $$X$$ is infinite and $$A \in \tau$$, we have two cases: $$A=\emptyset$$ and then $$\overline{A}=\emptyset$$ too, or $$A$$ is cofinite, hence infinite and the only closed set containing $$A$$ is $$X$$ (as an infinite set cannot be a subset of a finite set), so then $$\overline{A}=X$$, indeed.

I suppose $$X$$ is infinite here. Note that the way you defined it, $$\tau$$ is not a topology. You need to add the empty set to it.

Now let's assume $$A$$ is an open set in that topology. If $$A$$ is empty then it is its own closure, that's trivial. Otherwise, the complement of $$A$$ is a finite set, hence $$A$$ itself must be infinite, and of course that implies that it can't be contained in any finite set. So the only closed set which contains $$A$$ is $$X$$ itself, hence $$X$$ is the closure.

• yes, sorry there is empty set, I will correct. Apr 13, 2019 at 19:43