Limit Points in Closed Extension Space/Subset
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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 $U \subseteq S^*_p$ 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 $U$.
$\blacksquare$
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: $21$