Clopen Sets in Finite Complement Topology
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Theorem
Let $T = \struct {S, \tau}$ be a finite complement topology on an infinite set $S$.
Then the only clopen sets of $T$ are $S$ and $\O$.
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.
$\blacksquare$
Sources
- 1964: Steven A. Gaal: Point Set Topology ... (previous) ... (next): Chapter $\text {I}$: Topological Spaces: $1$. Open Sets and Closed Sets