Closure of Open Set of 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 $T$.
Then $U^- = S^*_p$ where $U^-$ denotes the closure of $U$ in $T^*_p$.
Proof
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 $U^-$.
$\blacksquare$
Sources
- 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text {II}$: Counterexamples: $12$. Closed Extension Topology: $21$