T3 1/2 Space is not necessarily T2 Space
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Theorem
Let $T = \struct {S, \tau}$ be a be a $T_{3 \frac 1 2}$ space.
Then it is not necessarily the case that $T$ is a $T_2$ (Hausdorff) space.
Proof
Let $S$ be a set and let $\PP$ be a partition on $S$ which is specifically not the (trivial) partition of singletons.
Let $T = \struct {S, \tau}$ be the partition space whose basis is $\PP$.
From Partition Topology is $T_{3 \frac 1 2}$, we have that $T$ is a $T_{3 \frac 1 2}$ space.
From Partition Topology is not Hausdorff, $T$ is not a $T_2$ (Hausdorff) space.
The result follows.
$\blacksquare$
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