Limit Points in Closed Extension Space

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Theorem

Let $T = \struct {S, \tau}$ be a topological space.

Let $T^*_p = \struct {S^*_p, \tau^*_p}$ be the closed extension space of $T$.


Let $x \in S$.

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


Similarly, let $U \subseteq S^*_p$ such that $p \in U$.

Let $x \in S$.


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


Proof

Every open set of $T^*_p = \struct {S^*_p, \tau^*_p}$ 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 $U$

and:

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

$\blacksquare$


Sources