Limit Points of Indiscrete Space

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Theorem

Let $T = \left({S, \left\{{\varnothing, S}\right\}}\right)$ be an indiscrete topological space consisting of at least two points.

Let $H$ be a subset of $T$ such that $H \ne \varnothing$.


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


Proof

By definition, $x \in \left({S, \tau}\right)$ is a limit point of $H$ if every open set $U \in \tau$ such that $x \in U$ contains some point of $H$ other than $x$.

Here, of course, there is only one open set that contains any points at all, and that is $S$.

So as $S$ contains more than one point, it follows that every point of $T$ is a limit point of $H$.

$\blacksquare$


Sources