Closure of Open Set of Closed Extension Space

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

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

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


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

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 $U^-$.