Fortissimo Space is not Separable

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Theorem

Let $T = \struct {S, \tau_p}$ be the fortissimo space on an uncountable set $S$.


Then $T$ is not a separable space.


Proof

Let $U$ be a countable subset of $S$.

By the definition of the fortissimo space, $U$ is closed.

From Closed Set Equals its Closure, $U^- = U \ne S$.

Thus, by definition, $U$ is not everywhere dense in $T$.

Thus, there exists no countable subset of $S$ which is everywhere dense in $T$.

So, by definition, $T$ is not a separable space.

$\blacksquare$


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