Fort Space is not Extremally Disconnected

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Theorem

Let $T = \left({S, \tau_p}\right)$ be a Fort space on an infinite set $S$.


Then $T$ is not an extremally disconnected space.


Proof

We note from Fort Space is Completely Normal that $T$ is a $T_2$ (Hausdorff) space.

Let $H \subseteq S$ be an infinite subset of $S$ such that $p \in \complement_S \left({H}\right)$ and $\complement_S \left({H}\right)$ is infinite.

Then $H$ is open as $p \in \complement_S \left({H}\right)$.

From Limit Points in Fort Space, as $H$ is infinite, $p$ is the only limit point of $H$.

So $H^- = H \cup \left\{{p}\right\}$ where $H^-$ is the closure of $H$.

But as $\complement_S \left({H^-}\right)$ is infinite and does not contain $p$, it follows that $H^-$ is not open in $T$.

So by definition $T$ is not an extremally disconnected space.

$\blacksquare$


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