Fort Space is Totally Separated

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Theorem

Let $T = \left({S, \tau_p}\right)$ 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 $\left\{{a}\right\}$ is closed, $\complement_S \left({\left\{{a}\right\}}\right)$ is open in $T$.

So we have a partition $\left\{{a}\right\} \mid \complement_S \left({\left\{{a}\right\}}\right)$ of $S$ such that $a \in \left\{{a}\right\}, b \in \complement_S \left({\left\{{a}\right\}}\right)$.

So, by definition $T$ is a totally separated space.

$\blacksquare$


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