Limit Points of Sequence in Indiscrete Space on Uncountable Set

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Theorem

Let $S$ be an uncountable set.

Let $T = \left({S, \left\{{\varnothing, S}\right\}}\right)$ be the indiscrete topological space on $S$.

Let $\left \langle {s_k} \right \rangle$ be a sequence in $T$.


Then every sequence in $T$ has an uncountable number of limit points.


Proof

From Sequence in Indiscrete Space converges to Every Point, $\left \langle {s_k} \right \rangle$ converges to every point of $S$.


As $S$ is uncountable, the result follows.

$\blacksquare$


Sources