# Limit Points of Infinite Subset of Finite Complement Space

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## Theorem

Let $T = \left({S, \tau}\right)$ be a finite complement space.

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

Then every point of $S$ is a limit point of $H$.

## Proof

Let $U \in \tau$ be any open set of $T$.

From Infinite Subset of Finite Complement Space Intersects Open Sets, we have that $U \cap H$ is infinite if and only if $H$ is infinite.

Let $x \in S$.

Then every open set $U$ in $T$ such that $x \in U$ also contains an infinite number of points of $H$ other than $x$.

Thus, by definition, $x$ is a limit point of $H$.

$\blacksquare$

## Sources

- 1970: Lynn Arthur Steen and J. Arthur Seebach, Jr.:
*Counterexamples in Topology*... (previous) ... (next): $\text{II}: \ 18 - 19: \ 1$