Existence of Topological Space which satisfies no Separation Axioms but T3

Theorem

There exists at least one example of a topological space for which none of the Tychonoff separation axioms are satisfied except for the $T_3$ axiom.

Proof

Proof by Counterexample:

Let $T$ be the topological space consisting of the double pointed topology on the Tychonoff corkscrew topology.

From Double Pointed Tychonoff Corkscrew fulfils no Separation Axioms but $T_3$, we have that $T$ satisfies none of the Tychonoff separation axioms except for the $T_3$ 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