Countable Complement Space is Irreducible
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Theorem
Let $T = \struct {S, \tau}$ be a countable complement topology on an uncountable set $S$.
Then $T$ is an irreducible space.
Proof
Let $U_1, U_2 \in \tau$ be non-empty open sets of $T$.
We have that both $U_1$ and $U_2$ are both uncountable if $S$ is.
From Uncountable Subset of Countable Complement Space Intersects Open Sets, they intersect each other.
Hence the result from definition of irreducible space.
$\blacksquare$
Sources
- 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text {II}$: Counterexamples: $20$. Countable Complement Topology: $4$