Fort Space is Totally Separated

Theorem

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

Then $T$ is a totally separated space.

Proof

Let $a, b \in S$ such that $a \ne b$.

From Clopen Points in Fort Space, all points in $S$ apart from $p$ are both open and closed in $T$.

Without loss of generality, suppose that $a \ne p$.

Then as $\set a$ is closed, $\relcomp S {\set a}$ is open in $T$.

So we have a partition $\set a \mid \relcomp S {\set a}$ of $S$ such that $a \in \set a, b \in \relcomp S {\set a}$.

So, by definition $T$ is a totally separated 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: $7$