Closed Extension Topology is not Hausdorff
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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 $T^*_p$ is not a $T_2$ (Hausdorf) space.
Proof
Aiming for a contradiction, suppose $T^*_p$ is not a $T_2$ (Hausdorf) space.
From $T_2$ Space is $T_1$ Space, $T^*_p$ is a $T_1$ (Fréchet) space.
But this contradicts Closed Extension Topology is not $T_1$
Hence by Proof by Contradiction $T^*_p$ can not be a $T_2$ (Hausdorf) space.
$\blacksquare$
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: $21$