Limit Points of Sequence in Indiscrete Space on Uncountable Set

Theorem

Let $S$ be an uncountable set.

Let $T = \struct {S, \set {\O, S} }$ be the indiscrete topological space on $S$.

Let $\sequence {s_n}$ be a sequence in $T$.

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

Proof

From Sequence in Indiscrete Space converges to Every Point, $\sequence {s_n}$ converges to every point of $S$.

As $S$ is uncountable, the result follows.

$\blacksquare$

Sources

• 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text {II}$: Counterexamples: $4$. Indiscrete Topology: $4$