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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?

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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.

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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.

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  • $\begingroup$ yes, sorry there is empty set, I will correct. $\endgroup$ – Joãonani Apr 13 '19 at 19:43

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