Closure of Infinite Subset of Finite Complement Space

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.

$\blacksquare$


Sources

  • 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text {II}$: Counterexamples: $18 \text { - } 19$. Finite Complement Topology: $1$
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