Partition Topology is not T0

Theorem
Let $S$ be a set and let $\mathcal P$ be a partition on $S$ which is not the (trivial) partition of singletons.

Let $T = \left({S, \vartheta}\right)$ be the partition space whose basis is $\mathcal P$.

Then $T$ is not a $T_0$ (Kolmogorov) space.

Thus $T$ is not $T_1$ (Fréchet), $T_2$ (Hausdorff) or $T_{2 \frac 1 2}$ (completely Hausdorff) either.

Proof
As $\mathcal P$ is not the partition of singletons, there exists some $H \in \mathcal P$ such that $a, b \in H: a \ne b$.

Any union of sets from $\mathcal P$ which includes $H$ will therefore contain both $a$ and $b$.

Therefore, any element of $\vartheta$ containing $a$ will also contain $b$, and similarly, any element of $\vartheta$ containing $b$ will also contain $a$.

Sothere is no open set in $T$ containing $a$ and not $b$, or $b$ and not $a$.

Hence the result, by definition of $T_0$ (Kolmogorov) space.

The assertions about $T$ not being $T_1$ (Fréchet), $T_2$ (Hausdorff) or $T_{2 \frac 1 2}$ (completely Hausdorff) are shown in Sequence of Implications of Separation Axioms.

Note the reason for the exception in the statement as given.

If $\mathcal P$ is the partition of singletons:
 * $\mathcal P = \left\{{\left\{{x}\right\}: x \in S}\right\}$

then $T$ as constructed is the discrete topology on $S$.

From Discrete Space satisfies all Separation Properties, in that case $T$ does satisfy the $T_0$ property.