Double Pointed Topology is not T0

Theorem
Let $T_1 = \left({S, \tau_S}\right)$ be a topological space.

Let $D = \left({A, \left\{ {\varnothing, A}\right\} }\right)$ be the indiscrete topology on an arbitrary doubleton $A = \left\{{a, b}\right\}$.

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

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