Separation Axioms on Double Pointed Topology/T3.5 Axiom

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 a $T_{3 \frac 1 2}$ space iff $T$ is also a $T_{3 \frac 1 2}$ space.