# Limit Points of Sequence in Indiscrete Space on Uncountable Set

## 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.

$\blacksquare$