Limit Points in Particular Point Space

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:

Particular Point Topology is Closed Extension Topology of Discrete Topology
Limit Points in Closed Extension Space

$\blacksquare$