Fort Space is Scattered/Proof 2

Theorem

Let $T = \struct {S, \tau_p}$ be a Fort space on an infinite set $S$.

Then $T$ is a scattered space.

Proof

Suppose that $H \subseteq T$ has no isolated points of $H$.

So, by definition, $H$ is dense in itself.

We have that:

a Fort Space is $T_1$

a Dense-in-itself Subset of $T_1$ Space is Infinite.

So $H$ is infinite, and so contains more than one point.

So $\exists q \in H: q \ne p$.

But, from Clopen Points in Fort Space, $\set q$ is open in $T$.

So $H \cap \set q = \set q$ and so by definition $q$ is isolated in $H$.

From this contradiction it follows that $H$ is not dense in itself and contains at least one 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: $23 \text { - } 24$. Fort Space: $8$