Clopen Points in Fort Space

Theorem

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

Let $q \in S: q \ne p$.

Then $\set q$ is both open and closed in $T$.

$\set p$ itself, on the other hand, is closed but not open.

Proof

We have that $\set q$ is finite so $\relcomp S {\set q}$ is cofinite.

So $\relcomp S {\set q}$ is open and so $\set q$ is closed.

Then we have that $p \notin \set q$ so $\set q$ is open.

However, $p \notin \relcomp S {\set p}$ and $\relcomp S {\set p}$ is infinite.

So $\set p$ is not open in $S$.

But $\set p$ is finite, so $\relcomp S {\set p}$ is cofinite.

Hence, as for $\set q$, we have that $\set p$ is closed.

$\blacksquare$

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: $7$