Modified Fort Space is Scattered

Theorem

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

Then $T$ is scattered.

Proof

We have that a modified Fort space is $T_1$.

We also have that a dense-in-itself subset of a $T_1$ space is infinite.

But from Isolated Points in Subsets of Modified Fort Space, we have that any subset of $T$ with more than two points has at least one isolated point.

So any dense-in-itself subset of $T$ must have an isolated point.

Hence the result, by definition of scattered 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: $6$