Closure in Infinite Particular Point Space is not Compact

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Theorem

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

Let $A \in \tau_p$ be open in $T$.

Let $A^-$ be the closure of $A$.


Then $A^-$ is not compact.


Proof

From Closure of Open Set of Particular Point Space, we have that $A^- = S$.

The result follows from Infinite Particular Point Space is not Compact.

$\blacksquare$


Sources