Finite Complement Space is Irreducible

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Theorem

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


Then $T$ is irreducible.


Proof

Let $U_1, U_2 \in \tau$ be non-empty open sets of $T$.

We have that $U_1$ and $U_2$ are both infinite if $S$ is.

From Infinite Subset of Finite Complement Space Intersects Open Sets, they intersect each other.

Hence the result from definition of irreducible space.

$\blacksquare$


Sources