Double Pointed Topology is not T0/Proof 2

Proof
By definition, the double pointed topology $\tau$ on $T_1$ is the product topology on $T_1 \times D$.

By definition, $D$ is the indiscrete space on a doubleton.

$T$ is a $T_0$ (Kolmogorov) space.

Then from Product Space is $T_0$ iff Factor Spaces are $T_0$ it follows that $D$ is also a $T_0$ (Kolmogorov) space.

But from Indiscrete Non-Singleton Space is not $T_0$, $D$ is not a $T_0$ (Kolmogorov) space.

The result follows by Proof by Contradiction.