Separation Axioms on Double Pointed Topology

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

Let $D = \left({\left\{{a, b}\right\}, \vartheta}\right)$ be the indiscrete topology on two points.

Let $T \times D$ be the double pointed topology on $T$.

Then:
 * $T \times D$ is not a $T_0$ (Kolmogorov) space, a $T_1$ (Fréchet) space, a $T_2$ (Hausdorff) space or a $T_{2 \frac 1 2}$ (completely Hausdorff) space.


 * $T \times D$ is a $T_3$ space, a $T_{3 \frac 1 2}$ space, a $T_4$ space or a $T_5$ space iff $T$ is.

$T_0$, $T_1$, $T_2$ and $T_{2 \frac 1 2}$ Axioms
From:
 * Double Pointed Topology is Not $T_0$
 * $T_1$ Space is $T_0$ Space
 * $T_2$ Space is $T_1$ Space
 * Completely Hausdorff Space is $T_2$ Space

we have that $T \times D$ is not $T_0$, $T_1$ or $T_2$ or $T_{2 \frac 1 2}$.