Limit Points in Uncountable Fort Space

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Theorem

Let $T = \left({S, \tau_p}\right)$ be an uncountable Fort space.

Let $U \subseteq S$ be a countably infinite subset of $S$.


Then $p$ is the only limit point of $U$.


Proof

Suppose $y \in S, y \ne p$.

We have by definition of Fort space that $\left\{{y}\right\}$ is open in $T$.

So there is no $z \in \left\{{y}\right\}: z \ne y, z \in U$.

Hence $y$ can not be a limit point of $U$.



Sources