Partition Topology is not Completely Hausdorff
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Theorem
Let $S$ be a set and let $\PP$ be a partition on $S$ which is not the (trivial) partition of singletons.
Let $T = \struct {S, \tau}$ be the partition space whose basis is $\PP$.
Then $T$ is not a $T_{2 \frac 1 2}$ (completely Hausdorff) space.
Proof
Aiming for a contradiction, suppose $T$ is a $T_{2 \frac 1 2}$ (completely Hausdorff) space.
Then from Completely Hausdorff Space is Hausdorff Space, $T$ is a $T_2$ (Hausdorff) space.
This contradicts the result Partition Topology is not Hausdorff.
Hence the result, by Proof by Contradiction.
$\blacksquare$
Sources
- 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text {II}$: Counterexamples: $5$. Partition Topology: $2$