# Limit Points in Particular Point Space

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## Theorem

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

Let $x \in S$ such that $x \ne p$.

Then $x$ is a limit point of $p$.

### Limit Points in Subset

Let $U \subseteq S$ such that $p \in U$.

Let $x \in S$ such that $x \ne p$.

Then $x$ is a limit point of $U$.

## Proof 1

Let $U \in \tau$ be an open set of $T$.

Then by definition of $T$:

- $p \in U$

That is, $U$ contains (at least) one point of $S$ which is distinct from $x$.

As $U$ is arbitrary, it follows that all open set of $T$ have this property.

The result follows by definition of the limit point of a point.

$\blacksquare$

## Proof 2

Follows directly from:

$\blacksquare$

## 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: $2$