Separation Axioms on Double Pointed Topology/T3.5 Axiom

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