Limit Points in Particular Point Space/Proof 1

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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 $p$.


Proof

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.

$\blacksquare$