Closed Sets of Closed Extension Topology
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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 the closed sets of $T^*_p$ (apart from $S^*_p$) are the closed sets of $T$.
This explains why $\tau^*_p$ is called the closed extension topology of $\tau$.
Proof
By definition:
- $\tau^*_p = \set {U \cup \set p: U \in \tau} \cup \set \O$
Let $V \subseteq S^*_p$ be closed in $T^*_p$.
Then $S^*_p \setminus V$ is open in $T^*_p$.
Then $\struct {S^*_p \setminus V} \setminus \set p$ is open in $T$.
From Set Difference with Union we have:
- $\struct {S^*_p \setminus V} \setminus \set p = S^*_p \setminus \struct {V \cup \set p}$
Hence the result.
$\blacksquare$
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: $20$