Existence of Topological Space which satisfies no Separation Axioms but T4

Theorem

There exists at least one example of a topological space for which none of the Tychonoff separation axioms are satisfied except for the $T_4$ axiom.

Proof

Let $T = \left({S, \tau}\right)$ be a countable complement topology on an uncountable set $S$.

Let $D = \left({\left\{{0, 1}\right\}, \vartheta}\right)$ be the indiscrete topology on two points.

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

Let $\left({T \times D}\right)^*_{\bar p}$ be the open extension topology on $S \times \left\{{0, 1}\right\} \cup \left\{{p}\right\}$ where $p \notin S \times \left\{{0, 1}\right\}$.

From Open Extension of Double Pointed Countable Complement Topology is $T_4$ Space, we have that $\left({T \times D}\right)^*_{\bar p}$ satisfies none of the Tychonoff separation axioms except for the $T_4$ axiom.

$\blacksquare$

Sources

• 1970: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology ... (previous) ... (next): $\text{I}: \ \S 2$