Closure of Infinite Subset of Finite Complement Space

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Theorem

Let $T = \struct {S, \tau}$ be a finite complement space.

Let $H \subseteq S$ be an infinite subset of $S$.


Then $H^- = S$ where $H^-$ is the closure of $S$.


Proof

Let $H$ be an infinite subset of $S$.

From Limit Points of Infinite Subset of Finite Complement Space, every point of $S$ is a limit point of $H$.

Hence the result from the definition of closure.

$\blacksquare$


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