Particular Point Space is Separable

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Theorem

Let $T = \left({S, \tau_p}\right)$ be a particular point space.


Then $T$ is separable.


Proof

By definition, $T$ is separable if and only if there exists a countable subset of $S$ which is everywhere dense in $T$.


Consider $U := \left\{{p}\right\} \subseteq S$.

By definition, $U$ is open in $T$.

From Closure of Open Set of Particular Point Space we have that $U^- = S$, where $U^-$ is the closure of $U$.

By definition, $U$ is everywhere dense in $T$.

$U$ is (trivially) countable.

Hence the result, by definition of a separable space.

$\blacksquare$


Sources