[Math] Dense set in an infinite cofinite space

Question

Let $X$ be an infinite cofinite space and let $A \subset X$ be infinite. Then $\overline A = X$

I am having a tough time in trying to understand this. Why is every infinite set in an infinite cofinite space dense ?

Answer 1

Let $x \in X$. Let $O$ be an open set that contains $x$. This means that $X \setminus O$ is finite. If $O$ does not intersect $A$, then $A \subseteq X \setminus O$, which is finite and as $A$ is infinite, this cannot happen. So $O$ intersects $A$. As $O$ was an arbitary neighbourhood of $x$, $x \in \overline{A}$. As $x$ was arbitrary, $\overline{A} = X$.

Another way to look at it: the only closed sets of $X$ are the finite ones and $X$. As $\overline{A}$ is a closed set that contains $A$, $\overline{A}$ cannot be finite, so it must be $X$.