Closure of Open Set of Closed Extension Space

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Theorem

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

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


Then $U^- = S^*_p$ where $U^-$ denotes the closure of $U$ in $T^*_p$.


Proof

By definition, $\forall U \in \tau^*_p, u \ne \varnothing: p \in U$.

From Limit Points in Closed Extension Space, every point in $S^*_p$ is a limit point of $p$.

So by definition of closure, every point in $S^*_p$ is in $U^-$.

$\blacksquare$


Sources