Existence of Topological Space which satisfies no Separation Axioms but T0

Theorem

There exists at least one example of a topological space for which none of the Tychonoff separation axioms are satisfied except for the $T_0$ (Kolmogorov) axiom.

Proof

Let $T$ be the overlapping interval space.

From Overlapping Interval Space fulfils no Separation Axioms but $T_0$, we have that $T$ satisfies none of the Tychonoff separation axioms except for the $T_0$ (Kolmogorov) axiom.

$\blacksquare$

Sources

• 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text I$: Basic Definitions: Section $2$: Separation Axioms