# Limit Points in Particular Point Space

## Theorem

Let $T = \left({S, \tau_p}\right)$ be a particular point space.

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

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

Similarly, 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

Follows directly from:

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

$\blacksquare$