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In here https://proofwiki.org/wiki/Fortissimo_Space_is_T1, it was proved that the uncountable set $X$ with the Fortissimo topology is $T_1$. So, every singleton is closed.

On the other hand, every point point in $X$ except $p$ is a isolated point. So, every set $\{x\}$, $x \in X-\{p\}$ is a clopen set. Is it true?

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    Yes, $T_1$ guarantees that points are closed and $x$ is isolated if and only if $\{x\}$ is open. – 2017-02-21
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    The Fortissimo space is just the one-point Lindelöf-ication of an uncountable discrete subspace $X \setminus \{p\}$. – 2017-02-21

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