Limit Points in Particular Point Space/Subset/Proof 2

Theorem

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

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 Subset of Closed Extension Space

$\blacksquare$