Existence of Completely Normal Space which is not Perfectly Normal
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Theorem
There exists at least one example of a completely normal topological space which is not perfectly normal.
Proof
Let $T$ be an uncountable Fort space.
From Fort Space is Completely Normal, $T$ is a completely normal space.
From Uncountable Fort Space is not Perfectly Normal, $T$ is not a perfectly normal space.
Hence the result.
$\blacksquare$
Sources
- 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text I$: Basic Definitions: Section $2$: Separation Axioms: Additional Separation Properties