Limit Points in Particular Point Space/Subset/Proof 1
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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
Every open set of $T = \struct {S, \tau_p}$ except $\O$ contains the point $p$ by definition.
So every open set $U \in \tau_p$ such that $x \in U$ contains $p$.
So by definition of the limit point of a set, $x$ is a limit point of $U$.
$\blacksquare$