# Existence of Topological Space which satisfies no Separation Axioms

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## Theorem

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

## Proof

Let $T$ be the topological space consisting of the double pointed topology on the countable complement topology on an uncountable set.

From Double Pointed Countable Complement Topology fulfils no Separation Axioms, we have that $T$ satisfies none of the Tychonoff separation axioms.

$\blacksquare$

## 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