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 $T = \struct {S, \tau}$ be a countable complement topology on an uncountable 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 $T$.

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

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