Double Pointed Topology is not T0/Proof 2

Theorem

Let $T_1 = \struct {S, \tau_S}$ be a topological space.

Let $D = \struct {A, \set {\O, A} }$ be the indiscrete topology on an arbitrary doubleton $A = \set {a, b}$.

Let $T = \struct {T_1 \times D, \tau}$ be the double pointed topological space on $T_1$.

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

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.

Aiming for a contradiction, suppose $T$ 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.

$\blacksquare$