Convergent Sequence in Particular Point Space

Theorem

Let $T = \struct {S, \tau_p}$ be a particular point space.

Let $\sequence {a_i}$ be a convergent sequence in $T$.

Except for a finite number of indices, the terms of $\sequence {a_i}$ for which $a_i \ne p$ are all equal.

Proof

Let $\sequence {a_i}$ be a convergent sequence in $T$ whose limit is $\alpha$.

Then by definition every open set in $T$ containing $\alpha$ contains all but a finite number of terms of $\sequence {a_i}$.

This includes the open set $\set {\alpha, p}$.

Thus $\sequence {a_i}$ is such that, except for a finite number of terms, all are equal either to $\alpha$ or $p$.

Thus $\sequence {a_i}$ is such that, except for a finite number of terms, all such that $a_i \ne p$, are equal to $\alpha$.

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: $8 \text { - } 10$. Particular Point Topology: $1$