Sequence in Indiscrete Space converges to Every Point

Theorem

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

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

Then $\sequence {s_n}$ converges to every point of $S$.

Proof

Let $\alpha \in S$.

By definition, $\sequence {s_n}$ converges to $\alpha$ if every open set in $T$ containing $\alpha$ contains all but a finite number of terms of $\sequence {s_n}$.

But as $T$ has only one open set containing any points at all, every point of $\sequence {s_n}$ is contained in every open set in $T$ containing $\alpha$.

Hence the result.

$\blacksquare$

Sources

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