Modified Fort Space is not T2

Theorem

Let $T = \struct {S, \tau_{a, b} }$ be a modified Fort space.

Then $T$ is not a $T_2$ (Hausdorff) space.

Proof

Consider $U, V \in \tau_{a, b}$ such that $a \in U, b \in V$.

We have that both $U$ and $V$ are cofinite.

So $U$ and $V$ must be infinite.

Suppose $U \cap V = \O$.

Then from Intersection with Complement is Empty iff Subset it follows that $U \subseteq \relcomp S V$ and so $U$ is finite.

But this contradicts the fact that $U$ is infinite.

So $U \cap V \ne \O$.

Hence the result by definition of $T_2$ (Hausdorff) space.

$\blacksquare$

Sources

• 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text {II}$: Counterexamples: $27$. Modified Fort Space: $3$