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Let $X$ be an infinite set and $T$ a topology on $X$ such that each infinite subset of $X$ is closed. Show that $T$ is the discrete topology.

The idea I have is to take two infinite subsets $A$, $B$. These are closed iff $X\setminus A$ is open and $X \setminus B$ is open. Then I want to see if $X \setminus A$ intersection $X \setminus B$ results in a singleton set. But I have problems in order to form the set $B$ properly to make the intersection described above, a singleton set.

Could you please give some suggestion?

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It suffices to show each singleton set $\{x\}$ is open... which is equivalent to showing that $X\setminus \{x\}$ is closed... which is true because...

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