Existence of Compact Space which Satisfies No Separation Axioms

Theorem
There exists at least one example of a compact space for which none of the Tychonoff separation axioms are satisfied.

Proof
Let $T = \struct {S, \tau}$ be a finite complement topology on an infinite set $S$.

Let $D = \struct {\set {0, 1}, \vartheta}$ be the indiscrete topology on two points.

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

From Double Pointed Finite Complement Topology is Compact, $T \times D$ is compact.

From Double Pointed Finite Complement Topology fulfils no Separation Axioms, $T \times D$ is not a $T_0$ space, $T_1$ space, $T_2$ space, $T_3$ space, $T_4$ space or $T_5$ space.