# Clopen Points in Modified Fort Space

## Theorem

Let $T = \left({S, \tau_{a, b}}\right)$ be a modified Fort space.

Then all points in $S \setminus \left\{{a, b}\right\}$ are both open and closed in $T$.

$a$ and $b$ themselves are not open in $T$, but they are closed in $T$.

## Proof

Let $p \in S: p \notin \left\{{a, b}\right\}$.

From the definition of modified Fort space, any subset of $S \setminus \left\{{a, b}\right\}$ is open in $T$.

It follows directly that as $p \in S \setminus \left\{{a, b}\right\}$ we have that $\left\{{p}\right\} \subseteq S \setminus \left\{{a, b}\right\}$.

Hence $p$ is open in $T$.

As for $a$ and $b$, we have that $S \setminus \left\{{a}\right\}$ and $S \setminus \left\{{b}\right\}$ are not finite and so $a$ and $b$ are not open in $T$.

$\Box$

For all points $p \in S$ (including $a$ and $b$), we have that $\left\{{p}\right\}$ is (trivially) finite.

It follows that while $S \setminus \left\{{p}\right\}$ contains either $a$ or $b$, it is cofinite.

Thus $S \setminus \left\{{p}\right\}$ is open in $T$.

It follows by definition that $p$ is closed in $T$.

$\blacksquare$

## Sources

- 1970: Lynn Arthur Steen and J. Arthur Seebach, Jr.:
*Counterexamples in Topology*... (previous) ... (next): $\text{II}: \ 27: \ 2$