Limit Points in Particular Point Space

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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

$\blacksquare$


Sources