Double Pointed Topology is not T0

Theorem
Let $T = \left({S, \vartheta}\right)$ be a double pointed topological space.

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

Proof
Let $T = S_1 \times S_2$ where:
 * $S_1$ is any arbitrary topological space
 * $S_2$ is the indiscrete topology on the doubleton $\left\{{a, b}\right\}$.

Let $x \in S_1$, and consider the point $\left({x, a}\right) \in S_1 \times S_2$.

Then:
 * $\forall U: \left({x, a}\right) \in U \implies \left({x, b}\right) \in U$
 * $\forall U: \left({x, b}\right) \in U \implies \left({x, a}\right) \in U$

as $S_2$ is the indiscrete topology.

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