Clopen Sets in Finite Complement Topology

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Theorem

Let $T = \left({S, \tau}\right)$ be a finite complement topology on an infinite set $S$.


Then the only clopen sets of $T$ are $S$ and $\varnothing$.


Proof

Let $U \in \tau$ be open in $T$.

Then by definition of finite complement topology, $S \setminus U$ is finite.

By definition of open set, $S \setminus U$ is closed.

As $S$ is infinite, it follows that $U$ must also be infinite.

Thus unless $U = S$, $S \setminus U$ can not be open.

Hence the result.

$\blacksquare$


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