Limit Points in Open Extension Space

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Theorem

Let $T = \left({S, \tau}\right)$ be a topological space.

Let $T^*_{\bar p} = \left({S^*_p, \tau^*_{\bar p}}\right)$ be the open extension space of $T$.


Let $x \in S$.

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


Similarly, let $U \subseteq S^*_p$.

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


Proof

Every open set of $T^*_p = \left({S^*_p, \tau^*_{\bar p}}\right)$ except $S^*_p$ does not contain the point $p$ by definition.

So every open set $U \in \tau^*_{\bar p}$ such that $p \in U$ (there is only the one such open set) contains $x$.

So:

by definition of the limit point of a set, $p$ is a limit point of $U$

and:

by definition of the limit point of a point, $p$ is a limit point of $x$.

$\blacksquare$


Sources