Accumulation Points for Sequence in Particular Point Space

Theorem

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

Let $\sequence {a_i}$ be an infinite sequence in $T$.

Let $\beta$ be an accumulation point of $\sequence {a_i}$.

Then $\beta$ is such that an infinite number of terms of $\sequence {a_i}$ are equal either to $\beta$ or to $p$.

Proof

Let $\beta$ be an accumulation point of $\sequence {a_i}$.

Then by definition:

all open sets of $T$ which contain $\beta$ also contain an infinite number of terms of $\sequence {a_i}$.

This condition applies to the open set $\set {\beta, p}$.

So $\set {\beta, p}$ contains an infinite number of terms of $\sequence {a_i}$.

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: $1$