Existence of T4 Space which is not T3 1/2
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Theorem
There exists at least one example of a topological space which is a $T_4$ space, but is not also a $T_{3 \frac 1 2}$ space.
Proof
Let $T$ be a Hjalmar Ekdal space.
From Hjalmar Ekdal Space is $T_4$, $T$ is a $T_4$ space.
From Hjalmar Ekdal Space is not $T_{3 \frac 1 2}$ Space, $T$ is not a $T_{3 \frac 1 2}$ 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: Completely Regular Spaces